Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is πr²h, right? And the volume of a sphere is (4/3)πr³. Let me confirm that... Yeah, I think that's correct. So the beaker is a cylinder with height 8 cm and radius 3 cm. The spherical tank has a radius of 6 cm.

First, maybe I should calculate the volume of one beaker. Let me do that.

For the cylinder: radius r = 3 cm, height h = 8 cm. So volume V_cylinder = π*(3)²*8. Let me compute that. 3 squared is 9, times 8 is 72. So V_cylinder = 72π cm³.

Now the sphere: radius R = 6 cm. Volume V_sphere = (4/3)π*(6)³. Let me compute that. 6 cubed is 6*6*6 = 216. Then multiply by 4/3: (4/3)*216. Let me do that step by step. 216 divided by 3 is 72, times 4 is 288. So V_sphere = 288π cm³.

So now I need to find how many beakers (each with volume 72π) are needed to fill the sphere (which has volume 288π). So the number of beakers would be the volume of the sphere divided by the volume of one beaker.

So that's 288π divided by 72π. The π cancels out, and 288 divided by 72 is 4. Wait, so does that mean it takes 4 beakers?

Wait a second, that seems too straightforward. Let me double-check my calculations.

Volume of cylinder: πr²h. r=3, h=8. 3 squared is 9, 9*8 is 72. Yep, 72π.

Volume of sphere: (4/3)πr³. r=6, so 6³ is 216. Multiply by 4/3: 216*(4/3) = 288. Yep, 288π.

Divide 288π by 72π: 288/72 = 4. So 4 beakers?

But intuitively, the sphere has a radius twice as big as the cylinder's radius. Wait, the cylinder's radius is 3 cm, and the sphere's radius is 6 cm. So the sphere's radius is double. But the cylinder's height is 8 cm.

But maybe the answer is indeed 4? Let me think again.

Alternatively, maybe there's some trick here. Like, maybe the beakers can't be filled completely or something? But the question says "how many such beakers of water will it take to fill a spherical tank". So it's assuming that you can pour the water from the beakers into the tank, and the tank is completely filled. So maybe it's just a matter of dividing the volumes.

But let me check the numbers again.

Sphere volume: (4/3)π*(6)^3. Let me compute that again. 6^3 is 216. 216*(4/3). 216 divided by 3 is 72, times 4 is 288. So yes, 288π.

Cylinder volume: π*(3)^2*8. 3^2 is 9, times 8 is 72. So 72π.

So 288 divided by 72 is 4. So 4 beakers.

Wait, but intuitively, a sphere with radius 6 cm... Let me think of the cylinder. The cylinder has radius 3 cm, same as the sphere's radius divided by 2. But the height is 8 cm.

Wait, maybe the sphere can fit into the cylinder? Or something. But the question isn't about fitting, it's about filling.

Alternatively, perhaps the question is trying to trick me into thinking about the volume, but maybe the beakers are not being poured directly into the spherical tank, but something else? But the wording says "to fill a spherical tank of radius 6 cm". So I think it's just volume.

Alternatively, maybe I made an error in the formula for the sphere? Let me check.

Yes, the volume of a sphere is (4/3)πr³. Yes.

So if the sphere has radius 6, then volume is 4/3 * π * 216 = 288π.

Yes. And the cylinder is 72π.

288 divided by 72 is 4. So 4 beakers.

But let me think again. If the radius of the sphere is 6, which is double the radius of the cylinder. But the height of the cylinder is 8 cm, which is less than the diameter of the sphere. The diameter of the sphere is 12 cm. So the cylinder is shorter than the sphere's diameter.

But volume-wise, the sphere is 288π, and each cylinder is 72π. So 288 divided by 72 is 4. So that would mean 4 beakers.

Alternatively, maybe I need to consider something else? Maybe the beakers are being arranged in some way? Like, stacking them inside the sphere? But the problem doesn't mention anything about stacking or arrangement. It just says "how many such beakers of water will it take to fill a spherical tank". So that implies that you are transferring the water from the beakers into the tank, so the total volume of the beakers must equal the volume of the tank.

Therefore, the answer is 4.

But wait, let me check once more.

Let me compute 72π * 4. 72*4 = 288. So yes, 288π. Which is exactly the volume of the sphere. So that seems correct.

Alternatively, maybe the problem is in US units? No, the problem is given in centimeters.

Wait, is there any chance that the question is referring to the beaker's volume when it's full? Like, maybe the beaker is filled to some level? But the question says "how many such beakers of water will it take to fill a spherical tank", so I think it's assuming that each beaker is completely filled and then poured into the tank.

Therefore, I think my answer is correct.

But maybe I made a mistake in calculation. Let me check once again.

Volume of sphere: (4/3)πr³. r=6. So:

First, 6³ is 6*6*6 = 216.

Multiply by 4/3: 216*(4/3) = (216/3)*4 = 72*4 = 288.

Yes.

Volume of cylinder: πr²h. r=3. So:

3²=9. h=8. 9*8=72. So π*72.

Yes.

So 288 divided by 72 is 4. Therefore, 4 beakers.

Hmm. So maybe the answer is 4.

But why does the problem give the height of the beaker? Maybe to trick me into thinking about something else?

Wait, maybe there's a mistake here. Let me think again. Suppose I confused diameter and radius? Let me check.

No, the beaker is a cylinder with radius 3 cm. The sphere has radius 6 cm. So the sphere's radius is double the cylinder's radius.

Alternatively, if someone thought that the sphere's diameter is the same as the cylinder's height? But no, the sphere's diameter is 12 cm, which is bigger than the cylinder's height of 8 cm.

Alternatively, maybe the question is expecting me to calculate something else?

Alternatively, maybe the beaker is a different shape? But the question says cylindrical beaker, so it's a cylinder.

Alternatively, maybe the problem is expecting the number of beakers needed if you stack them inside the sphere? But then you have to think about how many cylinders can fit into a sphere. But that's a different problem, and it's more complicated, involving packing efficiency or something. But the question doesn't mention anything about packing, just about filling the tank with water from the beakers.

Therefore, unless there's some trick, the answer should be 4.

But maybe the problem is from an exam where the answer is different? Wait, maybe I made a mistake in the formula. Wait, let me check the formula for the volume of a sphere again.

Yes, the volume of a sphere is (4/3)πr³. For example, if radius is 1, volume is 4/3 π. If radius is 2, volume is (4/3)π*(8) = 32/3 π.

So, if radius is 6, volume is (4/3)π*(6)^3 = (4/3)π*216 = 288π. Correct.

Cylinder: πr²h. r=3, h=8. So π*9*8 = 72π. Correct.

So 288π divided by 72π is 4.

So the answer is 4.

Alternatively, maybe the question is in liters? But no, the units are cm, so volume is in cm³.

Alternatively, maybe the tank is not completely filled? No, the question says "to fill a spherical tank", so it's completely filled.

Therefore, I think my answer is correct.

Wait, but just to make sure, maybe the question is a trick question? For example, if you have 4 beakers, each of volume 72π, then total water is 288π, which is exactly the volume of the sphere. Therefore, you need 4 beakers.

Therefore, the answer is 4.

I think that is correct. Maybe I was overcomplicating it, but the answer is 4.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers of water are needed to fill a spherical tank, we start by computing the **volumes** of both the beaker and the tank.

---

### **Volume of a Cylinder (Beaker)**

The volume $ V $ of a cylinder is given by the formula:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

Substitute the values:

$$
V_{\text{cylinder}} = \pi \cdot 3^2 \cdot 8 = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### **Volume of a Sphere (Tank)**

The volume $ V $ of a sphere is given by:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

Substitute the values:

$$
V_{\text{sphere}} = \frac{4}{3} \pi \cdot 6^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### **Determine Number of Beakers Required**

We now divide the total volume of the tank by the volume of one beaker to find how many beakers are needed:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### **Conclusion**

Thus, it takes exactly **4 cylindrical beakers** of water to completely fill a spherical tank with a radius of 6 cm.

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is πr²h, right? And the volume of a sphere is (4/3)πr³. Let me confirm that... Yeah, I think that's correct. So the beaker is a cylinder with height 8 cm and radius 3 cm. The spherical tank has a radius of 6 cm.

First, maybe I should calculate the volume of one beaker. Let me do that.

For the cylinder: radius r = 3 cm, height h = 8 cm. So volume V_cylinder = π*(3)²*8. Let me compute that. 3 squared is 9, times 8 is 72. So V_cylinder = 72π cm³.

Now the sphere: radius R = 6 cm. Volume V_sphere = (4/3)π*(6)³. Let me compute that. 6 cubed is 6*6*6 = 216. Then multiply by 4/3: (4/3)*216. Let me do that step by step. 216 divided by 3 is 72, times 4 is 288. So V_sphere = 288π cm³.

So now I need to find how many beakers (each with volume 72π) are needed to fill the sphere (which has volume 288π). So the number of beakers would be the volume of the sphere divided by the volume of one beaker.

So that's 288π divided by 72π. The π cancels out, and 288 divided by 72 is 4. Wait, so does that mean it takes 4 beakers?

Wait a second, that seems too straightforward. Let me double-check my calculations.

Volume of cylinder: πr²h. r=3, h=8. 3 squared is 9, 9*8 is 72. Yep, 72π.

Volume of sphere: (4/3)πr³. r=6, so 6³ is 216. Multiply by 4/3: 216*(4/3) = 288. Yep, 288π.

Divide 288π by 72π: 288/72 = 4. So 4 beakers?

But intuitively, the sphere has a radius twice as big as the cylinder's radius. Wait, the cylinder's radius is 3 cm, and the sphere's radius is 6 cm. So the sphere's radius is double. But the cylinder's height is 8 cm.

But maybe the answer is indeed 4? Let me think again.

Alternatively, maybe there's some trick here. Like, maybe the beakers can't be filled completely or something? But the question says "how many such beakers of water will it take to fill a spherical tank". So it's assuming that you can pour the water from the beakers into the tank, and the tank is completely filled. So maybe it's just a matter of dividing the volumes.

But let me check the numbers again.

Sphere volume: (4/3)π*(6)^3. Let me compute that again. 6^3 is 216. 216*(4/3). 216 divided by 3 is 72, times 4 is 288. So yes, 288π.

Cylinder volume: π*(3)^2*8. 3^2 is 9, times 8 is 72. So 72π.

So 288 divided by 72 is 4. So 4 beakers.

Wait, but intuitively, a sphere with radius 6 cm... Let me think of the cylinder. The cylinder has radius 3 cm, same as the sphere's radius divided by 2. But the height is 8 cm.

Wait, maybe the sphere can fit into the cylinder? Or something. But the question isn't about fitting, it's about filling.

Alternatively, perhaps the question is trying to trick me into thinking about the volume, but maybe the beakers are not being poured directly into the spherical tank, but something else? But the wording says "to fill a spherical tank of radius 6 cm". So I think it's just volume.

Alternatively, maybe I made an error in the formula for the sphere? Let me check.

Yes, the volume of a sphere is (4/3)πr³. Yes.

So if the sphere has radius 6, then volume is 4/3 * π * 216 = 288π.

Yes. And the cylinder is 72π.

288 divided by 72 is 4. So 4 beakers.

But let me think again. If the radius of the sphere is 6, which is double the radius of the cylinder. But the height of the cylinder is 8 cm, which is less than the diameter of the sphere. The diameter of the sphere is 12 cm. So the cylinder is shorter than the sphere's diameter.

But volume-wise, the sphere is 288π, and each cylinder is 72π. So 288 divided by 72 is 4. So that would mean 4 beakers.

Alternatively, maybe I need to consider something else? Maybe the beakers are being arranged in some way? Like, stacking them inside the sphere? But the problem doesn't mention anything about stacking or arrangement. It just says "how many such beakers of water will it take to fill a spherical tank". So that implies that you are transferring the water from the beakers into the tank, so the total volume of the beakers must equal the volume of the tank.

Therefore, the answer is 4.

But wait, let me check once more.

Let me compute 72π * 4. 72*4 = 288. So yes, 288π. Which is exactly the volume of the sphere. So that seems correct.

Alternatively, maybe the problem is in US units? No, the problem is given in centimeters.

Wait, is there any chance that the question is referring to the beaker's volume when it's full? Like, maybe the beaker is filled to some level? But the question says "how many such beakers of water will it take to fill a spherical tank", so I think it's assuming that each beaker is completely filled and then poured into the tank.

Therefore, I think my answer is correct.

But maybe I made a mistake in calculation. Let me check once again.

Volume of sphere: (4/3)πr³. r=6. So:

First, 6³ is 6*6*6 = 216.

Multiply by 4/3: 216*(4/3) = (216/3)*4 = 72*4 = 288.

Yes.

Volume of cylinder: πr²h. r=3. So:

3²=9. h=8. 9*8=72. So π*72.

Yes.

So 288 divided by 72 is 4. Therefore, 4 beakers.

Hmm. So maybe the answer is 4.

But why does the problem give the height of the beaker? Maybe to trick me into thinking about something else?

Wait, maybe there's a mistake here. Let me think again. Suppose I confused diameter and radius? Let me check.

No, the beaker is a cylinder with radius 3 cm. The sphere has radius 6 cm. So the sphere's radius is double the cylinder's radius.

Alternatively, if someone thought that the sphere's diameter is the same as the cylinder's height? But no, the sphere's diameter is 12 cm, which is bigger than the cylinder's height of 8 cm.

Alternatively, maybe the question is expecting me to calculate something else?

Alternatively, maybe the beaker is a different shape? But the question says cylindrical beaker, so it's a cylinder.

Alternatively, maybe the problem is expecting the number of beakers needed if you stack them inside the sphere? But then you have to think about how many cylinders can fit into a sphere. But that's a different problem, and it's more complicated, involving packing efficiency or something. But the question doesn't mention anything about packing, just about filling the tank with water from the beakers.

Therefore, unless there's some trick, the answer should be 4.

But maybe the problem is from an exam where the answer is different? Wait, maybe I made a mistake in the formula. Wait, let me check the formula for the volume of a sphere again.

Yes, the volume of a sphere is (4/3)πr³. For example, if radius is 1, volume is 4/3 π. If radius is 2, volume is (4/3)π*(8) = 32/3 π.

So, if radius is 6, volume is (4/3)π*(6)^3 = (4/3)π*216 = 288π. Correct.

Cylinder: πr²h. r=3, h=8. So π*9*8 = 72π. Correct.

So 288π divided by 72π is 4.

So the answer is 4.

Alternatively, maybe the question is in liters? But no, the units are cm, so volume is in cm³.

Alternatively, maybe the tank is not completely filled? No, the question says "to fill a spherical tank", so it's completely filled.

Therefore, I think my answer is correct.

Wait, but just to make sure, maybe the question is a trick question? For example, if you have 4 beakers, each of volume 72π, then total water is 288π, which is exactly the volume of the sphere. Therefore, you need 4 beakers.

Therefore, the answer is 4.

I think that is correct. Maybe I was overcomplicating it, but the answer is 4.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers of water are needed to fill a spherical tank, we start by computing the **volumes** of both the beaker and the tank.

---

### **Volume of a Cylinder (Beaker)**

The volume $ V $ of a cylinder is given by the formula:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

Substitute the values:

$$
V_{\text{cylinder}} = \pi \cdot 3^2 \cdot 8 = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### **Volume of a Sphere (Tank)**

The volume $ V $ of a sphere is given by:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

Substitute the values:

$$
V_{\text{sphere}} = \frac{4}{3} \pi \cdot 6^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### **Determine Number of Beakers Required**

We now divide the total volume of the tank by the volume of one beaker to find how many beakers are needed:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### **Conclusion**

Thus, it takes exactly **4 cylindrical beakers** of water to completely fill a spherical tank with a radius of 6 cm.

$$
\boxed{4}
$$

Solution: We first have to remember our formulas for the volumes of 3 dimensional objects. The volume of a cylinder with radius $r$ and height $h$ is $r^2h\pi$ and the volume of a sphere with radius $r$ is $\frac{4}{3} r^3 \pi$. Since the cylindrical beaker has a height of 8 centimeters and a radius of 3 centimeters, that means that its volume is $3^2\cdot8\cdot\pi=72\pi$ cubic centimeters. Since the sphere has a radius of 6 centimeters, its volume is $\frac{4}{3}\cdot6^3\pi = 288\pi$ cubic centimeters. The number of beakers of what it will take to fill the spherical tank is just the ratio of the volume of the tank to the volume of the cylinder, which is given by $\dfrac{288\pi}{72\pi}=\boxed{4}$.


Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, let me think. I remember there are some trigonometric identities for sine of sums and differences. Maybe I can use those to simplify the expression. Let me recall them.

First, the sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B. Similarly, the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So if I let A = arcsin 0.4 and B = arcsin 0.5, then the expression becomes sin(A + B) * sin(A - B). Wait, is that right? Let me check. Yes, because the first term is sin(A + B) and the second term is sin(A - B). So maybe I can compute each part separately and then multiply them together?

Alternatively, maybe there's a way to combine them. Let me think. If I have sin(A + B) * sin(A - B), that looks similar to some identity. Wait, there is an identity for sin(A + B) * sin(A - B). Let me recall. Alternatively, I can think of it as [sin(A + B)][sin(A - B)] which is equal to sin²A - sin²B. Is that correct? Let me verify. Let me expand both terms:

First, expand sin(A + B) * sin(A - B):

= [sin A cos B + cos A sin B] * [sin A cos B - cos A sin B]

That looks like (x + y)(x - y) where x = sin A cos B and y = cos A sin B. So that becomes x² - y².

Calculating x²: (sin A cos B)² = sin²A cos²B

Calculating y²: (cos A sin B)² = cos²A sin²B

Therefore, sin(A + B) * sin(A - B) = sin²A cos²B - cos²A sin²B. Hmm, is there a way to simplify this further?

Alternatively, maybe I can use other identities. Let me think. Alternatively, since I know sin(A + B) * sin(A - B) = sin²A - sin²B? Wait, let me check with specific values. Let me take A = 0, B = 0. Then sin(0 + 0) * sin(0 - 0) = 0 * 0 = 0. On the other hand, sin²0 - sin²0 = 0 - 0 = 0. Okay, that works. Let me try another example. Let A = π/2, B = 0. Then sin(π/2 + 0) * sin(π/2 - 0) = sin(π/2) * sin(π/2) = 1 * 1 = 1. On the other hand, sin²(π/2) - sin²0 = 1 - 0 = 1. That works too. Let me try another. Let A = π/4, B = π/4. Then sin(π/2) * sin(0) = 1 * 0 = 0. On the other hand, sin²(π/4) - sin²(π/4) = 0. So that also works. So maybe the identity is indeed sin(A + B) * sin(A - B) = sin²A - sin²B. Therefore, if that's the case, then I can use this identity to simplify the problem.

So if I can compute sin²A - sin²B, where A = arcsin 0.4 and B = arcsin 0.5, then that would give me the value of the expression. Let me check. Wait, let me verify this identity again. Let's expand sin(A + B) * sin(A - B) using the formula:

sin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

Multiplying them:

[sin A cos B]^2 - [cos A sin B]^2 = sin²A cos²B - cos²A sin²B. Hmm, so that's different from sin²A - sin²B. Wait, but in my previous examples, they were equal. Let me check with A = π/4 and B = π/4.

sin²A - sin²B is sin²(π/4) - sin²(π/4) = 0. On the other hand, sin²A cos²B - cos²A sin²B is (sin²A cos²B - cos²A sin²B). Since A = B, this is sin²A cos²A - cos²A sin²A = 0, which matches. So in that case, it's equal. Wait, but in general, is sin²A cos²B - cos²A sin²B equal to sin²A - sin²B? Let me test with A = π/2, B = 0. Then sin²A cos²B - cos²A sin²B = sin²(π/2) cos²(0) - cos²(π/2) sin²(0) = 1 * 1 - 0 * 0 = 1. On the other hand, sin²A - sin²B = 1 - 0 = 1. So that works. Wait, another example. Let A = π/6, B = π/6. Then sin²A cos²B - cos²A sin²B = sin²(π/6) cos²(π/6) - cos²(π/6) sin²(π/6) = 0. Which equals sin²A - sin²B = 0. So in these examples, it seems to hold. Wait, but is there a general identity?

Let me try to manipulate sin²A cos²B - cos²A sin²B. Let me factor:

sin²A cos²B - cos²A sin²B = sin²A cos²B - sin²B cos²A. Hmm, maybe factor something else? Let's factor out cos²A cos²B? Not sure. Alternatively, write it as sin²A cos²B - sin²B cos²A = cos²B sin²A - cos²A sin²B. Alternatively, factor sin²A - sin²B? Let me see:

If we factor, maybe using the identity sin²A - sin²B = sin(A + B) sin(A - B). Wait, that's the reverse of what we had earlier. Wait, actually, the identity is sin²A - sin²B = sin(A + B) sin(A - B). So if that is true, then sin(A + B) sin(A - B) = sin²A - sin²B. Which is the identity I thought earlier. So then, in our case, the expression sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4) is equal to sin²(arcsin 0.4) - sin²(arcsin 0.5). Wait, so that's 0.4² - 0.5². Because sin(arcsin x) is x. Therefore, this simplifies to 0.16 - 0.25 = -0.09. Therefore, the value of the entire expression is -0.09. Wait, so is that the answer? But let me verify this carefully because sometimes there can be mistakes in identities.

Let me confirm once again. If I have sin(A + B) * sin(A - B) = sin²A - sin²B. Let me check with A = 30°, B = 45°, just to be safe. Let me compute both sides numerically. Let me take A = π/6, B = π/4.

First, compute sin(A + B) * sin(A - B):

A + B = π/6 + π/4 = (2π + 3π)/12 = 5π/12. sin(5π/12) ≈ sin(75°) ≈ 0.9659.

A - B = π/6 - π/4 = (2π - 3π)/12 = -π/12. sin(-π/12) ≈ -0.2588.

So the product is approximately 0.9659 * (-0.2588) ≈ -0.25.

Now, sin²A - sin²B = sin²(π/6) - sin²(π/4) = (0.5)^2 - (√2/2)^2 = 0.25 - 0.5 = -0.25. So that matches. Therefore, the identity is correct. Therefore, in our problem, since A = arcsin(0.4), so sin(A) = 0.4, and B = arcsin(0.5), so sin(B) = 0.5. Then sin²A - sin²B = (0.4)^2 - (0.5)^2 = 0.16 - 0.25 = -0.09. Therefore, the value of the original expression is -0.09, which is -9/100. Therefore, the answer is -0.09. However, the question says to put the answer in boxed notation. But maybe they want a fraction? Let me check: 0.09 is 9/100, so -9/100. So maybe -9/100? Or maybe they want it as a decimal? Let me check if there's an alternative approach that might result in a different answer. Let me go through the steps again to be sure.

Alternatively, maybe I can compute the two terms separately and then multiply them. Let me try that method as cross-verification.

First, compute sin(arcsin 0.4 + arcsin 0.5). Let me denote θ = arcsin 0.4 and φ = arcsin 0.5. Then, we need to compute sin(θ + φ) and sin(φ - θ), then multiply them.

First, let's compute θ + φ. To compute sin(θ + φ), we can use the formula sinθ cosφ + cosθ sinφ. Similarly, sin(φ - θ) is sinφ cosθ - cosφ sinθ. Then multiplying them together:

[sinθ cosφ + cosθ sinφ][sinφ cosθ - cosφ sinθ] = same as earlier, which we know equals sin²θ - sin²φ. Therefore, same result. So if I compute this way, I need to compute sinθ, cosθ, sinφ, cosφ.

Given θ = arcsin 0.4, so sinθ = 0.4, so cosθ = sqrt(1 - sin²θ) = sqrt(1 - 0.16) = sqrt(0.84). Similarly, φ = arcsin 0.5, so sinφ = 0.5, cosφ = sqrt(1 - 0.25) = sqrt(0.75).

Therefore, let me compute sin(θ + φ) = sinθ cosφ + cosθ sinφ = 0.4 * sqrt(0.75) + sqrt(0.84) * 0.5.

Similarly, sin(φ - θ) = sinφ cosθ - cosφ sinθ = 0.5 * sqrt(0.84) - sqrt(0.75) * 0.4.

Now, if I compute the product [0.4 * sqrt(0.75) + sqrt(0.84) * 0.5] * [0.5 * sqrt(0.84) - 0.4 * sqrt(0.75)].

Let me compute each term step by step. Let me denote sqrt(0.75) as s and sqrt(0.84) as t.

Thus, the first term is 0.4s + 0.5t.

The second term is 0.5t - 0.4s.

Multiplying them: (0.4s + 0.5t)(0.5t - 0.4s) = (0.5t)^2 - (0.4s)^2 = 0.25t² - 0.16s².

Wait, that's the same as earlier! Because this is (x + y)(x - y) where x = 0.5t, y = 0.4s. Wait, no, actually, (a + b)(c - d) isn't a difference of squares unless there is some relation. Wait, but in this case, the first term is 0.4s + 0.5t and the second term is 0.5t - 0.4s. So that is (a + b)(b - a) where a = 0.4s, b = 0.5t. Which is equal to b² - a². Therefore, it's (0.5t)^2 - (0.4s)^2. Which is 0.25t² - 0.16s². Now, t is sqrt(0.84), so t² is 0.84. Similarly, s is sqrt(0.75), so s² is 0.75. Therefore:

0.25 * 0.84 - 0.16 * 0.75 = Let me compute each term:

0.25 * 0.84 = 0.21

0.16 * 0.75 = 0.12

Therefore, 0.21 - 0.12 = 0.09. Wait, but earlier I thought it was -0.09? Wait, hold on, in the previous approach I had sin²θ - sin²φ = 0.16 - 0.25 = -0.09, but here it's 0.25t² - 0.16s² = 0.21 - 0.12 = 0.09. Wait, there's a discrepancy here. Which one is correct?

Wait, in the first approach, using identity, I said that sin(A + B) * sin(A - B) = sin²A - sin²B. But here, in this calculation, I get 0.09 instead of -0.09. Therefore, there must be an error in my reasoning. Which one is correct?

Let me check with numbers. Let me compute the numerical value using the first method. Let me compute sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me compute these two terms numerically.

First, compute θ = arcsin(0.4). Let me calculate θ in radians. arcsin(0.4) ≈ 0.4115 radians. φ = arcsin(0.5) = π/6 ≈ 0.5236 radians.

Compute θ + φ ≈ 0.4115 + 0.5236 ≈ 0.9351 radians. Then sin(θ + φ) ≈ sin(0.9351). Let me compute this: sin(0.9351) ≈ approx. Let me check with calculator steps. Well, 0.9351 radians is approx 53.6 degrees (since π radians is 180, so 0.9351 * (180/π) ≈ 0.9351 * 57.3 ≈ 53.6 degrees). The sine of 53.6 degrees is approx 0.806 (since sin(53.13 degrees) is 0.8, which is arctan(4/3), so maybe close). Let me use calculator-like approximations. Alternatively, let me use exact expressions.

But maybe better to compute using actual numbers. Let me use more accurate values. Let me use a calculator for arcsin(0.4):

arcsin(0.4) ≈ 0.411516846 radians.

arcsin(0.5) = π/6 ≈ 0.5235987756 radians.

Adding them: θ + φ ≈ 0.411516846 + 0.5235987756 ≈ 0.9351156216 radians.

sin(θ + φ) ≈ sin(0.9351156216). Let me calculate this value. Using Taylor series or calculator-like approximation. Alternatively, use the formula:

sin(θ + φ) = sinθ cosφ + cosθ sinφ.

We know sinθ = 0.4, sinφ = 0.5.

cosθ = sqrt(1 - 0.4²) = sqrt(1 - 0.16) = sqrt(0.84) ≈ 0.916515139.

cosφ = sqrt(1 - 0.5²) = sqrt(0.75) ≈ 0.8660254038.

Therefore, sin(θ + φ) = 0.4 * 0.8660254038 + 0.916515139 * 0.5 ≈ 0.3464101615 + 0.4582575695 ≈ 0.804667731.

Now, compute sin(φ - θ). φ - θ ≈ 0.5235987756 - 0.411516846 ≈ 0.1120819296 radians. So sin(φ - θ) ≈ sin(0.1120819296). Let me calculate that using the formula sin(φ - θ) = sinφ cosθ - cosφ sinθ.

Which is 0.5 * 0.916515139 - 0.8660254038 * 0.4 ≈ 0.4582575695 - 0.3464101615 ≈ 0.111847408.

Therefore, multiplying the two results: 0.804667731 * 0.111847408 ≈ ?

Let me compute that:

First, approximate 0.8 * 0.11 = 0.088, but more accurately:

0.804667731 × 0.111847408 ≈ Let me do it step by step:

Multiply 0.804667731 × 0.111847408:

≈ 0.8 * 0.1118474 ≈ 0.08947792

Plus 0.004667731 × 0.1118474 ≈ approx 0.000521

Total approx 0.08947792 + 0.000521 ≈ 0.09. So approximately 0.09.

But earlier, using the identity, I thought it was -0.09. Wait, but according to this numerical calculation, it's positive 0.09. Therefore, there must be an error in my previous application of the identity.

Wait, so where is the mistake? Earlier, I thought sin(A + B) * sin(A - B) = sin²A - sin²B. But according to the numerical calculation, that would be 0.4² - 0.5² = 0.16 - 0.25 = -0.09. But the actual product is about +0.09. Therefore, there's a contradiction here. Which suggests that my identity was incorrect.

Wait, that is a big problem. So where is the mistake?

Let me go back. Let me derive sin(A + B) * sin(A - B) again. Let me write:

sin(A + B) * sin(A - B) = [sin A cos B + cos A sin B] * [sin A cos B - cos A sin B] = (sin A cos B)^2 - (cos A sin B)^2.

Which is sin²A cos²B - cos²A sin²B. Now, if I factor this expression:

sin²A cos²B - cos²A sin²B = sin²A (1 - sin²B) - (1 - sin²A) sin²B = sin²A - sin²A sin²B - sin²B + sin²A sin²B = sin²A - sin²B.

Wait! Wait, that simplifies to sin²A - sin²B. But according to my numerical calculation, it's positive 0.09, but according to this derivation, it should be sin²A - sin²B = 0.16 - 0.25 = -0.09. But my numerical calculation showed approximately 0.09. Therefore, there is a contradiction here. So where is the mistake?

Wait, in the numerical calculation, I have sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me check if I computed the second sine correctly. The second term is sin(arcsin 0.5 - arcsin 0.4). Which is sin(φ - θ). But φ = arcsin 0.5, θ = arcsin 0.4. So φ - θ is positive, since φ > θ? Wait, arcsin 0.5 is π/6 ≈ 0.5236, arcsin 0.4 ≈ 0.4115. Therefore, φ - θ ≈ 0.112, which is positive. Therefore, sin(φ - θ) is positive. Therefore, the product is [positive number] * [positive number]?

Wait, wait! Wait, in my calculation above, sin(arcsin 0.4 + arcsin 0.5) was approx 0.8046, which is positive, and sin(arcsin 0.5 - arcsin 0.4) ≈ 0.1118, also positive. Therefore, the product is positive. However, according to the identity, it's sin²A - sin²B = negative. Therefore, contradiction. Therefore, my derivation must have an error.

Wait, let me check my derivation again. Let me take:

sin(A + B) * sin(A - B) = sin²A cos²B - cos²A sin²B. Then, I tried to expand as sin²A(1 - sin²B) - (1 - sin²A) sin²B. Let me check that:

sin²A cos²B = sin²A (1 - sin²B). Correct. Similarly, cos²A sin²B = (1 - sin²A) sin²B. Correct. So then:

sin²A cos²B - cos²A sin²B = sin²A(1 - sin²B) - (1 - sin²A) sin²B = sin²A - sin²A sin²B - sin²B + sin²A sin²B = sin²A - sin²B. Therefore, that seems correct. Therefore, according to this derivation, it should be sin²A - sin²B. But according to the numerical example, it's the negative. So what is going on?

Wait, let me recalculate the numerical example. Wait, earlier when I used A = π/6, B = π/4, I had sin(A + B) * sin(A - B) ≈ -0.25 and sin²A - sin²B = 1 - 0.5 = 0.5? Wait no, no, in that case, A = π/6, B = π/4. So sin²A - sin²B = sin²(π/6) - sin²(π/4) = (1/2)^2 - (√2/2)^2 = 1/4 - 1/2 = -1/4 = -0.25. Which matched the numerical result. So that worked. So for that case, it was negative. But in our current problem, the numerical calculation suggests positive value, while the identity suggests negative. Therefore, there must be an error in my numerical calculation. Let me check again.

Let me recalculate the numerical value. So, θ = arcsin(0.4) ≈ 0.4115 radians, φ = arcsin(0.5) ≈ 0.5236 radians. Then:

Compute sin(θ + φ): approx sin(0.9351) ≈ 0.8046 (as before).

Compute sin(φ - θ): sin(φ - θ) = sin(0.1121) ≈ 0.1118 (as before). Multiply them: 0.8046 * 0.1118 ≈ let me compute more accurately.

Let me use exact decimal values:

sin(θ + φ) ≈ 0.804667731

sin(φ - θ) ≈ 0.111847408

Multiplying these two:

First, 0.8 * 0.11 = 0.088

Now compute 0.004667731 * 0.11 ≈ 0.00051345

Then, 0.8 * 0.001847408 ≈ 0.001477926

Then, 0.004667731 * 0.001847408 ≈ negligible, say ~0.00000863

So adding up: approx 0.088 + 0.00051345 + 0.001477926 + 0.00000863 ≈ 0.09. So approximately 0.09. However, according to identity, it should be sin²A - sin²B = 0.16 - 0.25 = -0.09. Which is negative. So there is a contradiction here. Therefore, I must have made a mistake in the identity.

Wait, but in the previous example with A = π/6 and B = π/4, the identity worked. Let me check that again. Let me recalculate:

θ = π/6, φ = π/4. A = θ, B = φ. Wait, no, in that case, we were calculating sin(A + B) * sin(A - B). Wait, but let me check what A and B are. If A = π/6 and B = π/4, then sin(A + B) * sin(A - B) = sin(7π/12) * sin(-π/12) = sin(7π/12) * (-sin(π/12)) ≈ 0.9659 * (-0.2588) ≈ -0.25. Which is equal to sin²A - sin²B = (sin²(π/6) - sin²(π/4)) = (0.25 - 0.5) = -0.25. So that worked. Therefore, the identity seems to hold. So why in our current problem, the numerical calculation contradicts the identity?

Wait, in this problem, the expression is sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Which is sin(θ + φ) * sin(φ - θ). Wait, in the previous example, it was sin(A + B) * sin(A - B). But here, it's sin(θ + φ) * sin(φ - θ). Is that the same as sin(A + B) * sin(A - B)? Yes, if A = θ and B = φ, then it's sin(A + B) * sin(B - A) = sin(A + B) * sin(-(A - B)) = -sin(A + B) * sin(A - B). Therefore, the expression is - [sin(A + B) * sin(A - B)]. Which would be - [sin²A - sin²B]. Therefore, in our case, the expression is equal to - [sin²A - sin²B] = sin²B - sin²A = 0.25 - 0.16 = 0.09. Which matches the numerical calculation. Ah! There was a mistake here. I had confused the order of the terms. The original expression is sin(A + B) * sin(B - A), which is sin(A + B)*sin(-(A - B)) = -sin(A + B)*sin(A - B). Therefore, the expression is equal to - [sin²A - sin²B] = sin²B - sin²A. Hence, 0.25 - 0.16 = 0.09, which matches the numerical result. Earlier, I had mistakenly thought the expression was sin(A + B)*sin(A - B), but actually, it is sin(A + B)*sin(B - A) which is equal to -sin(A + B)*sin(A - B). Therefore, the correct value is sin²B - sin²A = 0.25 - 0.16 = 0.09. Therefore, the answer is 0.09, not -0.09. My mistake earlier was not considering the sign from the difference in the second sine term. So this shows the importance of careful attention to signs.

Therefore, the correct value is 0.09, which is 9/100. Therefore, the answer is 0.09 or 9/100. Let me check once again.

Original problem: sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me denote the first term as sin(A + B) and the second as sin(B - A). Which is equal to sin(A + B) * sin(B - A) = sin(A + B) * [ - sin(A - B) ] = - sin(A + B) * sin(A - B). Then, using the identity sin(A + B) * sin(A - B) = sin²A - sin²B, so the entire expression is - (sin²A - sin²B) = sin²B - sin²A. Which is 0.5² - 0.4² = 0.25 - 0.16 = 0.09. Hence, correct.

Therefore, the answer is 0.09. Which is 9/100. So boxed answer is \boxed{0.09} or \boxed{\dfrac{9}{100}}. Depending on what's required. Since the question didn't specify, but 0.09 is straightforward. However, maybe they want a fraction. Let me check if 0.09 is 9/100. Yes, 9 divided by 100 is 0.09. So both are correct. In many math problems, fractions are preferred. However, since the original problem uses decimal numbers (0.4, 0.5), it's possible that decimal is acceptable. Let me check if there's a simpler way to write it. Alternatively, maybe the exact value is needed? Well, 0.09 is exact. However, to be precise, since 0.4 and 0.5 are exact decimals, the answer is exact as 0.09. Alternatively, 9/100 is also exact. Both are acceptable. Given that, and since the problem is in a context where LaTeX is used, maybe boxed as \boxed{0.09} is fine. However, sometimes problems prefer fractions. Let me check with the calculation using exact terms.

Alternatively, let me compute the exact value using the previous method. sin²B - sin²A = (0.5)^2 - (0.4)^2 = 0.25 - 0.16 = 0.09. Yes, exactly. So 0.09 is exact. Therefore, the answer is 0.09. Hence, boxed answer is \boxed{0.09}.

But just to be thorough, let me compute the numerical value using exact expressions:

Earlier, we had:

sin(θ + φ) ≈ 0.804667731

sin(φ - θ) ≈ 0.111847408

Multiplying them: 0.804667731 * 0.111847408 ≈ let me do this multiplication more precisely.

Multiply 0.804667731 × 0.111847408:

First, note that 0.8 × 0.1 = 0.08

Then, 0.004667731 × 0.111847408 ≈ approximately:

Let me compute 0.004667731 × 0.1 = 0.0004667731

0.004667731 × 0.011847408 ≈ approx 0.00005523

So total approx 0.0004667731 + 0.00005523 ≈ 0.000522. Then adding to the 0.08 gives approximately 0.080522. Wait, but earlier I thought it was about 0.09. Wait, maybe my previous approximation was wrong. Let me use more accurate multiplication:

Alternatively, use calculator-like steps. Let me write:

Let me take 0.804667731 × 0.111847408.

Let me note that 0.804667731 × 0.111847408 = (approx) ?

Alternatively, use approximate decimal values:

Let me take 0.8047 × 0.1118.

Compute 0.8 × 0.1118 = 0.08944

Compute 0.0047 × 0.1118 ≈ 0.000525

Adding together: 0.08944 + 0.000525 ≈ 0.089965 ≈ 0.09. So yes, approx 0.09. Therefore, the exact value is 0.09. Therefore, this confirms that the exact value is 0.09.

Therefore, the correct answer is 0.09. Hence, the original confusion was due to a sign error when applying the identity. Initially, I thought the expression was sin(A + B) * sin(A - B), but it was actually sin(A + B) * sin(B - A), which introduces a negative sign. Therefore, the correct value is 0.09.

**Final Answer**
\boxed{0.09}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we can simplify it using trigonometric identities and properties of inverse sine functions.

---

### Step 1: Define Angles

Let:

- $ A = \arcsin 0.4 $
- $ B = \arcsin 0.5 $

Then the expression becomes:

$$
\sin(A + B) \cdot \sin(B - A)
$$

---

### Step 2: Use the Sine Identity

We use the identity:

$$
\sin(A + B) \cdot \sin(B - A) = -\sin(A + B) \cdot \sin(A - B)
$$

This is because $ \sin(B - A) = -\sin(A - B) $. Hence:

$$
\sin(A + B) \cdot \sin(B - A) = -\sin(A + B) \cdot \sin(A - B)
$$

We now use the sum and difference identity:

$$
\sin(A + B) \cdot \sin(A - B) = \sin^2 A - \sin^2 B
$$

Therefore:

$$
\sin(A + B) \cdot \sin(B - A) = -(\sin^2 A - \sin^2 B) = \sin^2 B - \sin^2 A
$$

---

### Step 3: Plug in Known Values

We are given:

- $ \sin A = \sin(\arcsin 0.4) = 0.4 $
- $ \sin B = \sin(\arcsin 0.5) = 0.5 $

Then:

$$
\sin^2 A = (0.4)^2 = 0.16 \\
\sin^2 B = (0.5)^2 = 0.25
$$

So:

$$
\sin^2 B - \sin^2 A = 0.25 - 0.16 = 0.09
$$

---

### Final Answer

$$
\boxed{0.09}
$$

[Assistant]
Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, let me think. I remember there are some trigonometric identities for sine of sums and differences. Maybe I can use those to simplify the expression. Let me recall them.

First, the sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B. Similarly, the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So if I let A = arcsin 0.4 and B = arcsin 0.5, then the expression becomes sin(A + B) * sin(A - B). Wait, is that right? Let me check. Yes, because the first term is sin(A + B) and the second term is sin(A - B). So maybe I can compute each part separately and then multiply them together?

Alternatively, maybe there's a way to combine them. Let me think. If I have sin(A + B) * sin(A - B), that looks similar to some identity. Wait, there is an identity for sin(A + B) * sin(A - B). Let me recall. Alternatively, I can think of it as [sin(A + B)][sin(A - B)] which is equal to sin²A - sin²B. Is that correct? Let me verify. Let me expand both terms:

First, expand sin(A + B) * sin(A - B):

= [sin A cos B + cos A sin B] * [sin A cos B - cos A sin B]

That looks like (x + y)(x - y) where x = sin A cos B and y = cos A sin B. So that becomes x² - y².

Calculating x²: (sin A cos B)² = sin²A cos²B

Calculating y²: (cos A sin B)² = cos²A sin²B

Therefore, sin(A + B) * sin(A - B) = sin²A cos²B - cos²A sin²B. Hmm, is there a way to simplify this further?

Alternatively, maybe I can use other identities. Let me think. Alternatively, since I know sin(A + B) * sin(A - B) = sin²A - sin²B? Wait, let me check with specific values. Let me take A = 0, B = 0. Then sin(0 + 0) * sin(0 - 0) = 0 * 0 = 0. On the other hand, sin²0 - sin²0 = 0 - 0 = 0. Okay, that works. Let me try another example. Let A = π/2, B = 0. Then sin(π/2 + 0) * sin(π/2 - 0) = sin(π/2) * sin(π/2) = 1 * 1 = 1. On the other hand, sin²(π/2) - sin²0 = 1 - 0 = 1. That works too. Let me try another. Let A = π/4, B = π/4. Then sin(π/2) * sin(0) = 1 * 0 = 0. On the other hand, sin²(π/4) - sin²(π/4) = 0. So that also works. So maybe the identity is indeed sin(A + B) * sin(A - B) = sin²A - sin²B. Therefore, if that's the case, then I can use this identity to simplify the problem.

So if I can compute sin²A - sin²B, where A = arcsin 0.4 and B = arcsin 0.5, then that would give me the value of the expression. Let me check. Wait, let me verify this identity again. Let's expand sin(A + B) * sin(A - B) using the formula:

sin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

Multiplying them:

[sin A cos B]^2 - [cos A sin B]^2 = sin²A cos²B - cos²A sin²B. Hmm, so that's different from sin²A - sin²B. Wait, but in my previous examples, they were equal. Let me check with A = π/4 and B = π/4.

sin²A - sin²B is sin²(π/4) - sin²(π/4) = 0. On the other hand, sin²A cos²B - cos²A sin²B is (sin²A cos²B - cos²A sin²B). Since A = B, this is sin²A cos²A - cos²A sin²A = 0, which matches. So in that case, it's equal. Wait, but in general, is sin²A cos²B - cos²A sin²B equal to sin²A - sin²B? Let me test with A = π/2, B = 0. Then sin²A cos²B - cos²A sin²B = sin²(π/2) cos²(0) - cos²(π/2) sin²(0) = 1 * 1 - 0 * 0 = 1. On the other hand, sin²A - sin²B = 1 - 0 = 1. So that works. Wait, another example. Let A = π/6, B = π/6. Then sin²A cos²B - cos²A sin²B = sin²(π/6) cos²(π/6) - cos²(π/6) sin²(π/6) = 0. Which equals sin²A - sin²B = 0. So in these examples, it seems to hold. Wait, but is there a general identity?

Let me try to manipulate sin²A cos²B - cos²A sin²B. Let me factor:

sin²A cos²B - cos²A sin²B = sin²A cos²B - sin²B cos²A. Hmm, maybe factor something else? Let's factor out cos²A cos²B? Not sure. Alternatively, write it as sin²A cos²B - sin²B cos²A = cos²B sin²A - cos²A sin²B. Alternatively, factor sin²A - sin²B? Let me see:

If we factor, maybe using the identity sin²A - sin²B = sin(A + B) sin(A - B). Wait, that's the reverse of what we had earlier. Wait, actually, the identity is sin²A - sin²B = sin(A + B) sin(A - B). So if that is true, then sin(A + B) sin(A - B) = sin²A - sin²B. Which is the identity I thought earlier. So then, in our case, the expression sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4) is equal to sin²(arcsin 0.4) - sin²(arcsin 0.5). Wait, so that's 0.4² - 0.5². Because sin(arcsin x) is x. Therefore, this simplifies to 0.16 - 0.25 = -0.09. Therefore, the value of the entire expression is -0.09. Wait, so is that the answer? But let me verify this carefully because sometimes there can be mistakes in identities.

Let me confirm once again. If I have sin(A + B) * sin(A - B) = sin²A - sin²B. Let me check with A = 30°, B = 45°, just to be safe. Let me compute both sides numerically. Let me take A = π/6, B = π/4.

First, compute sin(A + B) * sin(A - B):

A + B = π/6 + π/4 = (2π + 3π)/12 = 5π/12. sin(5π/12) ≈ sin(75°) ≈ 0.9659.

A - B = π/6 - π/4 = (2π - 3π)/12 = -π/12. sin(-π/12) ≈ -0.2588.

So the product is approximately 0.9659 * (-0.2588) ≈ -0.25.

Now, sin²A - sin²B = sin²(π/6) - sin²(π/4) = (0.5)^2 - (√2/2)^2 = 0.25 - 0.5 = -0.25. So that matches. Therefore, the identity is correct. Therefore, in our problem, since A = arcsin(0.4), so sin(A) = 0.4, and B = arcsin(0.5), so sin(B) = 0.5. Then sin²A - sin²B = (0.4)^2 - (0.5)^2 = 0.16 - 0.25 = -0.09. Therefore, the value of the original expression is -0.09, which is -9/100. Therefore, the answer is -0.09. However, the question says to put the answer in boxed notation. But maybe they want a fraction? Let me check: 0.09 is 9/100, so -9/100. So maybe -9/100? Or maybe they want it as a decimal? Let me check if there's an alternative approach that might result in a different answer. Let me go through the steps again to be sure.

Alternatively, maybe I can compute the two terms separately and then multiply them. Let me try that method as cross-verification.

First, compute sin(arcsin 0.4 + arcsin 0.5). Let me denote θ = arcsin 0.4 and φ = arcsin 0.5. Then, we need to compute sin(θ + φ) and sin(φ - θ), then multiply them.

First, let's compute θ + φ. To compute sin(θ + φ), we can use the formula sinθ cosφ + cosθ sinφ. Similarly, sin(φ - θ) is sinφ cosθ - cosφ sinθ. Then multiplying them together:

[sinθ cosφ + cosθ sinφ][sinφ cosθ - cosφ sinθ] = same as earlier, which we know equals sin²θ - sin²φ. Therefore, same result. So if I compute this way, I need to compute sinθ, cosθ, sinφ, cosφ.

Given θ = arcsin 0.4, so sinθ = 0.4, so cosθ = sqrt(1 - sin²θ) = sqrt(1 - 0.16) = sqrt(0.84). Similarly, φ = arcsin 0.5, so sinφ = 0.5, cosφ = sqrt(1 - 0.25) = sqrt(0.75).

Therefore, let me compute sin(θ + φ) = sinθ cosφ + cosθ sinφ = 0.4 * sqrt(0.75) + sqrt(0.84) * 0.5.

Similarly, sin(φ - θ) = sinφ cosθ - cosφ sinθ = 0.5 * sqrt(0.84) - sqrt(0.75) * 0.4.

Now, if I compute the product [0.4 * sqrt(0.75) + sqrt(0.84) * 0.5] * [0.5 * sqrt(0.84) - 0.4 * sqrt(0.75)].

Let me compute each term step by step. Let me denote sqrt(0.75) as s and sqrt(0.84) as t.

Thus, the first term is 0.4s + 0.5t.

The second term is 0.5t - 0.4s.

Multiplying them: (0.4s + 0.5t)(0.5t - 0.4s) = (0.5t)^2 - (0.4s)^2 = 0.25t² - 0.16s².

Wait, that's the same as earlier! Because this is (x + y)(x - y) where x = 0.5t, y = 0.4s. Wait, no, actually, (a + b)(c - d) isn't a difference of squares unless there is some relation. Wait, but in this case, the first term is 0.4s + 0.5t and the second term is 0.5t - 0.4s. So that is (a + b)(b - a) where a = 0.4s, b = 0.5t. Which is equal to b² - a². Therefore, it's (0.5t)^2 - (0.4s)^2. Which is 0.25t² - 0.16s². Now, t is sqrt(0.84), so t² is 0.84. Similarly, s is sqrt(0.75), so s² is 0.75. Therefore:

0.25 * 0.84 - 0.16 * 0.75 = Let me compute each term:

0.25 * 0.84 = 0.21

0.16 * 0.75 = 0.12

Therefore, 0.21 - 0.12 = 0.09. Wait, but earlier I thought it was -0.09? Wait, hold on, in the previous approach I had sin²θ - sin²φ = 0.16 - 0.25 = -0.09, but here it's 0.25t² - 0.16s² = 0.21 - 0.12 = 0.09. Wait, there's a discrepancy here. Which one is correct?

Wait, in the first approach, using identity, I said that sin(A + B) * sin(A - B) = sin²A - sin²B. But here, in this calculation, I get 0.09 instead of -0.09. Therefore, there must be an error in my reasoning. Which one is correct?

Let me check with numbers. Let me compute the numerical value using the first method. Let me compute sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me compute these two terms numerically.

First, compute θ = arcsin(0.4). Let me calculate θ in radians. arcsin(0.4) ≈ 0.4115 radians. φ = arcsin(0.5) = π/6 ≈ 0.5236 radians.

Compute θ + φ ≈ 0.4115 + 0.5236 ≈ 0.9351 radians. Then sin(θ + φ) ≈ sin(0.9351). Let me compute this: sin(0.9351) ≈ approx. Let me check with calculator steps. Well, 0.9351 radians is approx 53.6 degrees (since π radians is 180, so 0.9351 * (180/π) ≈ 0.9351 * 57.3 ≈ 53.6 degrees). The sine of 53.6 degrees is approx 0.806 (since sin(53.13 degrees) is 0.8, which is arctan(4/3), so maybe close). Let me use calculator-like approximations. Alternatively, let me use exact expressions.

But maybe better to compute using actual numbers. Let me use more accurate values. Let me use a calculator for arcsin(0.4):

arcsin(0.4) ≈ 0.411516846 radians.

arcsin(0.5) = π/6 ≈ 0.5235987756 radians.

Adding them: θ + φ ≈ 0.411516846 + 0.5235987756 ≈ 0.9351156216 radians.

sin(θ + φ) ≈ sin(0.9351156216). Let me calculate this value. Using Taylor series or calculator-like approximation. Alternatively, use the formula:

sin(θ + φ) = sinθ cosφ + cosθ sinφ.

We know sinθ = 0.4, sinφ = 0.5.

cosθ = sqrt(1 - 0.4²) = sqrt(1 - 0.16) = sqrt(0.84) ≈ 0.916515139.

cosφ = sqrt(1 - 0.5²) = sqrt(0.75) ≈ 0.8660254038.

Therefore, sin(θ + φ) = 0.4 * 0.8660254038 + 0.916515139 * 0.5 ≈ 0.3464101615 + 0.4582575695 ≈ 0.804667731.

Now, compute sin(φ - θ). φ - θ ≈ 0.5235987756 - 0.411516846 ≈ 0.1120819296 radians. So sin(φ - θ) ≈ sin(0.1120819296). Let me calculate that using the formula sin(φ - θ) = sinφ cosθ - cosφ sinθ.

Which is 0.5 * 0.916515139 - 0.8660254038 * 0.4 ≈ 0.4582575695 - 0.3464101615 ≈ 0.111847408.

Therefore, multiplying the two results: 0.804667731 * 0.111847408 ≈ ?

Let me compute that:

First, approximate 0.8 * 0.11 = 0.088, but more accurately:

0.804667731 × 0.111847408 ≈ Let me do it step by step:

Multiply 0.804667731 × 0.111847408:

≈ 0.8 * 0.1118474 ≈ 0.08947792

Plus 0.004667731 × 0.1118474 ≈ approx 0.000521

Total approx 0.08947792 + 0.000521 ≈ 0.09. So approximately 0.09.

But earlier, using the identity, I thought it was -0.09. Wait, but according to this numerical calculation, it's positive 0.09. Therefore, there must be an error in my previous application of the identity.

Wait, so where is the mistake? Earlier, I thought sin(A + B) * sin(A - B) = sin²A - sin²B. But according to the numerical calculation, that would be 0.4² - 0.5² = 0.16 - 0.25 = -0.09. But the actual product is about +0.09. Therefore, there's a contradiction here. Which suggests that my identity was incorrect.

Wait, that is a big problem. So where is the mistake?

Let me go back. Let me derive sin(A + B) * sin(A - B) again. Let me write:

sin(A + B) * sin(A - B) = [sin A cos B + cos A sin B] * [sin A cos B - cos A sin B] = (sin A cos B)^2 - (cos A sin B)^2.

Which is sin²A cos²B - cos²A sin²B. Now, if I factor this expression:

sin²A cos²B - cos²A sin²B = sin²A (1 - sin²B) - (1 - sin²A) sin²B = sin²A - sin²A sin²B - sin²B + sin²A sin²B = sin²A - sin²B.

Wait! Wait, that simplifies to sin²A - sin²B. But according to my numerical calculation, it's positive 0.09, but according to this derivation, it should be sin²A - sin²B = 0.16 - 0.25 = -0.09. But my numerical calculation showed approximately 0.09. Therefore, there is a contradiction here. So where is the mistake?

Wait, in the numerical calculation, I have sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me check if I computed the second sine correctly. The second term is sin(arcsin 0.5 - arcsin 0.4). Which is sin(φ - θ). But φ = arcsin 0.5, θ = arcsin 0.4. So φ - θ is positive, since φ > θ? Wait, arcsin 0.5 is π/6 ≈ 0.5236, arcsin 0.4 ≈ 0.4115. Therefore, φ - θ ≈ 0.112, which is positive. Therefore, sin(φ - θ) is positive. Therefore, the product is [positive number] * [positive number]?

Wait, wait! Wait, in my calculation above, sin(arcsin 0.4 + arcsin 0.5) was approx 0.8046, which is positive, and sin(arcsin 0.5 - arcsin 0.4) ≈ 0.1118, also positive. Therefore, the product is positive. However, according to the identity, it's sin²A - sin²B = negative. Therefore, contradiction. Therefore, my derivation must have an error.

Wait, let me check my derivation again. Let me take:

sin(A + B) * sin(A - B) = sin²A cos²B - cos²A sin²B. Then, I tried to expand as sin²A(1 - sin²B) - (1 - sin²A) sin²B. Let me check that:

sin²A cos²B = sin²A (1 - sin²B). Correct. Similarly, cos²A sin²B = (1 - sin²A) sin²B. Correct. So then:

sin²A cos²B - cos²A sin²B = sin²A(1 - sin²B) - (1 - sin²A) sin²B = sin²A - sin²A sin²B - sin²B + sin²A sin²B = sin²A - sin²B. Therefore, that seems correct. Therefore, according to this derivation, it should be sin²A - sin²B. But according to the numerical example, it's the negative. So what is going on?

Wait, let me recalculate the numerical example. Wait, earlier when I used A = π/6, B = π/4, I had sin(A + B) * sin(A - B) ≈ -0.25 and sin²A - sin²B = 1 - 0.5 = 0.5? Wait no, no, in that case, A = π/6, B = π/4. So sin²A - sin²B = sin²(π/6) - sin²(π/4) = (1/2)^2 - (√2/2)^2 = 1/4 - 1/2 = -1/4 = -0.25. Which matched the numerical result. So that worked. So for that case, it was negative. But in our current problem, the numerical calculation suggests positive value, while the identity suggests negative. Therefore, there must be an error in my numerical calculation. Let me check again.

Let me recalculate the numerical value. So, θ = arcsin(0.4) ≈ 0.4115 radians, φ = arcsin(0.5) ≈ 0.5236 radians. Then:

Compute sin(θ + φ): approx sin(0.9351) ≈ 0.8046 (as before).

Compute sin(φ - θ): sin(φ - θ) = sin(0.1121) ≈ 0.1118 (as before). Multiply them: 0.8046 * 0.1118 ≈ let me compute more accurately.

Let me use exact decimal values:

sin(θ + φ) ≈ 0.804667731

sin(φ - θ) ≈ 0.111847408

Multiplying these two:

First, 0.8 * 0.11 = 0.088

Now compute 0.004667731 * 0.11 ≈ 0.00051345

Then, 0.8 * 0.001847408 ≈ 0.001477926

Then, 0.004667731 * 0.001847408 ≈ negligible, say ~0.00000863

So adding up: approx 0.088 + 0.00051345 + 0.001477926 + 0.00000863 ≈ 0.09. So approximately 0.09. However, according to identity, it should be sin²A - sin²B = 0.16 - 0.25 = -0.09. Which is negative. So there is a contradiction here. Therefore, I must have made a mistake in the identity.

Wait, but in the previous example with A = π/6 and B = π/4, the identity worked. Let me check that again. Let me recalculate:

θ = π/6, φ = π/4. A = θ, B = φ. Wait, no, in that case, we were calculating sin(A + B) * sin(A - B). Wait, but let me check what A and B are. If A = π/6 and B = π/4, then sin(A + B) * sin(A - B) = sin(7π/12) * sin(-π/12) = sin(7π/12) * (-sin(π/12)) ≈ 0.9659 * (-0.2588) ≈ -0.25. Which is equal to sin²A - sin²B = (sin²(π/6) - sin²(π/4)) = (0.25 - 0.5) = -0.25. So that worked. Therefore, the identity seems to hold. So why in our current problem, the numerical calculation contradicts the identity?

Wait, in this problem, the expression is sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Which is sin(θ + φ) * sin(φ - θ). Wait, in the previous example, it was sin(A + B) * sin(A - B). But here, it's sin(θ + φ) * sin(φ - θ). Is that the same as sin(A + B) * sin(A - B)? Yes, if A = θ and B = φ, then it's sin(A + B) * sin(B - A) = sin(A + B) * sin(-(A - B)) = -sin(A + B) * sin(A - B). Therefore, the expression is - [sin(A + B) * sin(A - B)]. Which would be - [sin²A - sin²B]. Therefore, in our case, the expression is equal to - [sin²A - sin²B] = sin²B - sin²A = 0.25 - 0.16 = 0.09. Which matches the numerical calculation. Ah! There was a mistake here. I had confused the order of the terms. The original expression is sin(A + B) * sin(B - A), which is sin(A + B)*sin(-(A - B)) = -sin(A + B)*sin(A - B). Therefore, the expression is equal to - [sin²A - sin²B] = sin²B - sin²A. Hence, 0.25 - 0.16 = 0.09, which matches the numerical result. Earlier, I had mistakenly thought the expression was sin(A + B)*sin(A - B), but actually, it is sin(A + B)*sin(B - A) which is equal to -sin(A + B)*sin(A - B). Therefore, the correct value is sin²B - sin²A = 0.25 - 0.16 = 0.09. Therefore, the answer is 0.09, not -0.09. My mistake earlier was not considering the sign from the difference in the second sine term. So this shows the importance of careful attention to signs.

Therefore, the correct value is 0.09, which is 9/100. Therefore, the answer is 0.09 or 9/100. Let me check once again.

Original problem: sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4). Let me denote the first term as sin(A + B) and the second as sin(B - A). Which is equal to sin(A + B) * sin(B - A) = sin(A + B) * [ - sin(A - B) ] = - sin(A + B) * sin(A - B). Then, using the identity sin(A + B) * sin(A - B) = sin²A - sin²B, so the entire expression is - (sin²A - sin²B) = sin²B - sin²A. Which is 0.5² - 0.4² = 0.25 - 0.16 = 0.09. Hence, correct.

Therefore, the answer is 0.09. Which is 9/100. So boxed answer is \boxed{0.09} or \boxed{\dfrac{9}{100}}. Depending on what's required. Since the question didn't specify, but 0.09 is straightforward. However, maybe they want a fraction. Let me check if 0.09 is 9/100. Yes, 9 divided by 100 is 0.09. So both are correct. In many math problems, fractions are preferred. However, since the original problem uses decimal numbers (0.4, 0.5), it's possible that decimal is acceptable. Let me check if there's a simpler way to write it. Alternatively, maybe the exact value is needed? Well, 0.09 is exact. However, to be precise, since 0.4 and 0.5 are exact decimals, the answer is exact as 0.09. Alternatively, 9/100 is also exact. Both are acceptable. Given that, and since the problem is in a context where LaTeX is used, maybe boxed as \boxed{0.09} is fine. However, sometimes problems prefer fractions. Let me check with the calculation using exact terms.

Alternatively, let me compute the exact value using the previous method. sin²B - sin²A = (0.5)^2 - (0.4)^2 = 0.25 - 0.16 = 0.09. Yes, exactly. So 0.09 is exact. Therefore, the answer is 0.09. Hence, boxed answer is \boxed{0.09}.

But just to be thorough, let me compute the numerical value using exact expressions:

Earlier, we had:

sin(θ + φ) ≈ 0.804667731

sin(φ - θ) ≈ 0.111847408

Multiplying them: 0.804667731 * 0.111847408 ≈ let me do this multiplication more precisely.

Multiply 0.804667731 × 0.111847408:

First, note that 0.8 × 0.1 = 0.08

Then, 0.004667731 × 0.111847408 ≈ approximately:

Let me compute 0.004667731 × 0.1 = 0.0004667731

0.004667731 × 0.011847408 ≈ approx 0.00005523

So total approx 0.0004667731 + 0.00005523 ≈ 0.000522. Then adding to the 0.08 gives approximately 0.080522. Wait, but earlier I thought it was about 0.09. Wait, maybe my previous approximation was wrong. Let me use more accurate multiplication:

Alternatively, use calculator-like steps. Let me write:

Let me take 0.804667731 × 0.111847408.

Let me note that 0.804667731 × 0.111847408 = (approx) ?

Alternatively, use approximate decimal values:

Let me take 0.8047 × 0.1118.

Compute 0.8 × 0.1118 = 0.08944

Compute 0.0047 × 0.1118 ≈ 0.000525

Adding together: 0.08944 + 0.000525 ≈ 0.089965 ≈ 0.09. So yes, approx 0.09. Therefore, the exact value is 0.09. Therefore, this confirms that the exact value is 0.09.

Therefore, the correct answer is 0.09. Hence, the original confusion was due to a sign error when applying the identity. Initially, I thought the expression was sin(A + B) * sin(A - B), but it was actually sin(A + B) * sin(B - A), which introduces a negative sign. Therefore, the correct value is 0.09.

**Final Answer**
\boxed{0.09}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we can simplify it using trigonometric identities and properties of inverse sine functions.

---

### Step 1: Define Angles

Let:

- $ A = \arcsin 0.4 $
- $ B = \arcsin 0.5 $

Then the expression becomes:

$$
\sin(A + B) \cdot \sin(B - A)
$$

---

### Step 2: Use the Sine Identity

We use the identity:

$$
\sin(A + B) \cdot \sin(B - A) = -\sin(A + B) \cdot \sin(A - B)
$$

This is because $ \sin(B - A) = -\sin(A - B) $. Hence:

$$
\sin(A + B) \cdot \sin(B - A) = -\sin(A + B) \cdot \sin(A - B)
$$

We now use the sum and difference identity:

$$
\sin(A + B) \cdot \sin(A - B) = \sin^2 A - \sin^2 B
$$

Therefore:

$$
\sin(A + B) \cdot \sin(B - A) = -(\sin^2 A - \sin^2 B) = \sin^2 B - \sin^2 A
$$

---

### Step 3: Plug in Known Values

We are given:

- $ \sin A = \sin(\arcsin 0.4) = 0.4 $
- $ \sin B = \sin(\arcsin 0.5) = 0.5 $

Then:

$$
\sin^2 A = (0.4)^2 = 0.16 \\
\sin^2 B = (0.5)^2 = 0.25
$$

So:

$$
\sin^2 B - \sin^2 A = 0.25 - 0.16 = 0.09
$$

---

### Final Answer

$$
\boxed{0.09}
$$

Solution:
From the angle addition and subtraction formulas,
\begin{align*}
\sin (x + y) &= \sin x \cos y + \cos x \sin y, \\
\sin (x - y) &= \sin x \cos y - \cos x \sin y,
\end{align*}so
\begin{align*}
\sin (x + y) \sin (x - y) &= (\sin x \cos y + \cos x \sin y)(\sin x \cos y - \cos x \sin y) \\
&= \sin^2 x \cos^2 y + \sin x \cos x \sin y \cos y - \sin x \cos x \sin y \cos y - \cos^2 x \sin^2 y \\
&= \sin^2 x (1 - \sin^2 y) - (1 - \sin^2 x) \sin^2 y \\
&= \sin^2 x - \sin^2 x \sin^2 y - \sin^2 y + \sin^2 x \sin^2 y \\
&= \sin^2 x - \sin^2 y.
\end{align*}Taking $x = \arcsin 0.5$ and $y = \arcsin 0.4,$ we get
\begin{align*}
\sin (\arcsin 0.5 + \arcsin 0.4) \cdot \sin (\arcsin 0.5 - \arcsin 0.4) &= \sin^2 (\arcsin 0.5) - \sin^2 (\arcsin 0.4) \\
&= 0.5^2 - 0.4^2 \\
&= 0.09 = \boxed{\frac{9}{100}}.
\end{align*}


Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. Letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, which is 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers.

Starting with the letters. The first thing that comes to mind is permutations because the order of the letters matters. For example, "AB" is different from "BA". Since the letters have to be different, I can't repeat any letters. So for the first letter, there are 26 choices. Once I pick the first letter, there are 25 remaining letters for the second position. So the number of possible combinations for the letters would be 26 multiplied by 25. Let me write that down:

Letters: 26 * 25

Wait, let me confirm that. If I have 26 choices for the first letter and then 25 for the second, yes, that's correct. Because after choosing the first letter, there are 25 left. Since the order matters here, it's a permutation problem. So permutations of 26 letters taken 2 at a time. The formula for permutations is P(n, k) = n! / (n - k)!. So for n=26 and k=2, that would be 26! / 24! Which is 26*25*24! / 24! = 26*25. Yep, same result. So that part is straightforward.

Now moving on to the numbers. The problem says two different one-digit numbers. So similar to the letters, but with digits. There are 10 digits from 0 to 9. Again, the order matters here because, for example, "12" is different from "21". So again, permutations. So for the first digit, there are 10 choices, and then for the second digit, there are 9 remaining choices. Therefore, the number of combinations for the numbers would be 10 * 9. Let me check that with the permutation formula as well. P(10, 2) = 10! / (10 - 2)! = 10! / 8! = 10*9*8! / 8! = 10*9. Yep, same result.

So now, to find the total number of license plates, I need to multiply the number of possible letter combinations by the number of possible number combinations. That would be:

Total = (26 * 25) * (10 * 9)

Let me calculate that. First, let me compute 26 * 25. 26 times 25. Well, 25*26. Let me think, 25*20 is 500, and 25*6 is 150, so total is 500 + 150 = 650. So 26*25 is 650. Then 10*9 is 90. So then total license plates are 650 * 90. Hmm, let me compute that.

Breaking down 650 * 90. Well, 650 * 90 is the same as 650 * 9 * 10. 650 * 9. Let me compute 650*9. 600*9 = 5400, and 50*9=450, so 5400 + 450 = 5850. Then multiplying by 10 gives 58,500. So total number of license plates is 58,500.

Wait, let me verify that multiplication again just to be sure. 650 multiplied by 90. Alternatively, 650 * 90 can be thought of as 65 * 10 * 9 * 10 = 65 * 9 * 100. 65 * 9 is 585. Then 585 * 100 is 58,500. Yep, same result. Okay, so that seems correct.

But let me just double-check all steps once again to ensure there are no mistakes.

First, letters: two different letters. Since order matters, permutations. 26P2 = 26*25 = 650. Correct.

Numbers: two different digits. Order matters, so 10P2 = 10*9 = 90. Correct.

Total combinations: 650 * 90 = 58,500. Correct.

Is there a possibility that the numbers can have leading zeros? For example, is "01" considered a valid two-digit number here? The problem says "two different one digit numbers". So each is a single digit, so the numbers are 0-9, and they just have to be different. So even if the first digit is zero, that's allowed. So for example, "01" is a valid combination. Therefore, the calculation for numbers is correct because we are considering all permutations, including those starting with zero. So no problem there.

Another thing to check: are the letters case-sensitive? The problem says "A through Z", so I assume it's uppercase only, and they are different. So no, there's no case sensitivity here. So no need to consider lowercase letters. So 26 letters total. Correct.

Are the digits allowed to be repeated? No, the problem says "two different one digit numbers", so digits must be different. Therefore, the calculation of 10*9 is correct.

So, all steps seem to check out. Therefore, the total number of license plates is 58,500.

But just to make sure, let me think of another way to approach this problem.

Alternative approach:

For the letters part: The first letter can be any of the 26 letters. The second letter can be any letter except the first one, so 25. So 26*25.

For the numbers part: Similarly, first digit can be 0-9 (10 options), second digit can be any digit except the first one, so 9. So 10*9.

Multiply them together: (26*25)*(10*9) = 650*90 = 58,500. Yep, same answer.

Alternatively, if I consider the entire license plate as a sequence of 4 characters, where the first two are letters and the last two are numbers, with all the constraints. But in that case, it's still breaking it down into two separate parts.

Another check: what if the numbers were allowed to repeat? Then it would be 10*10 = 100, but since they have to be different, it's 10*9 = 90. Similarly for letters.

Another check: if the letters were allowed to repeat, then it would be 26*26 = 676, but since they have to be different, it's 26*25 = 650.

So all these checks are consistent.

Alternatively, if I calculate the total possibilities without considering the restrictions, then subtract the ones that violate the restrictions, but that would be more complicated. Let me see:

Total number of license plates with 2 letters and 2 numbers, allowing repeats: 26*26*10*10 = 676*100 = 67,600. Then subtract the ones with repeated letters or repeated numbers. But this is inclusion-exclusion, which is more complex.

But since the problem specifies that the letters are different and the numbers are different, the first method of calculating permutations for each part is straightforward and accurate.

Alternatively, maybe someone could think that the two different letters could be arranged in some way, but since we already considered permutations, that's accounted for.

So, I think my answer is correct. Therefore, the total number of license plates is 58,500.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed, we analyze the structure: each license plate must consist of **2 different letters followed by 2 different digits**, where:

- Letters are selected from the English alphabet (A to Z).
- Digits are selected from the set of digits 0 to 9.

---

### Step 1: Calculate the number of valid letter combinations

We are choosing **2 different letters** from 26 letters (A–Z) **in order**, because the license plate has a specific sequence.

Since the letters must be **different** and **order matters**, this is a **permutation** problem.

$$
\text{Number of letter combinations} = P(26, 2) = \frac{26!}{(26 - 2)!} = 26 \times 25 = 650
$$

---

### Step 2: Calculate the number of valid digit combinations

Similarly, the two **digits must be different** and **order matters**, so again, it's a permutation of 10 digits taken 2 at a time.

$$
\text{Number of digit combinations} = P(10, 2) = \frac{10!}{(10 - 2)!} = 10 \times 9 = 90
$$

---

### Step 3: Multiply the combinations for total number of license plates

Each combination of 2 letters can be paired with any of the valid combinations of 2 digits (since the choices are independent), so we multiply the two values:

$$
\text{Total license plates} = 650 \times 90 = 58{,}500
$$

---

### Final Answer

$$
\boxed{58500}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. Letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, which is 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers.

Starting with the letters. The first thing that comes to mind is permutations because the order of the letters matters. For example, "AB" is different from "BA". Since the letters have to be different, I can't repeat any letters. So for the first letter, there are 26 choices. Once I pick the first letter, there are 25 remaining letters for the second position. So the number of possible combinations for the letters would be 26 multiplied by 25. Let me write that down:

Letters: 26 * 25

Wait, let me confirm that. If I have 26 choices for the first letter and then 25 for the second, yes, that's correct. Because after choosing the first letter, there are 25 left. Since the order matters here, it's a permutation problem. So permutations of 26 letters taken 2 at a time. The formula for permutations is P(n, k) = n! / (n - k)!. So for n=26 and k=2, that would be 26! / 24! Which is 26*25*24! / 24! = 26*25. Yep, same result. So that part is straightforward.

Now moving on to the numbers. The problem says two different one-digit numbers. So similar to the letters, but with digits. There are 10 digits from 0 to 9. Again, the order matters here because, for example, "12" is different from "21". So again, permutations. So for the first digit, there are 10 choices, and then for the second digit, there are 9 remaining choices. Therefore, the number of combinations for the numbers would be 10 * 9. Let me check that with the permutation formula as well. P(10, 2) = 10! / (10 - 2)! = 10! / 8! = 10*9*8! / 8! = 10*9. Yep, same result.

So now, to find the total number of license plates, I need to multiply the number of possible letter combinations by the number of possible number combinations. That would be:

Total = (26 * 25) * (10 * 9)

Let me calculate that. First, let me compute 26 * 25. 26 times 25. Well, 25*26. Let me think, 25*20 is 500, and 25*6 is 150, so total is 500 + 150 = 650. So 26*25 is 650. Then 10*9 is 90. So then total license plates are 650 * 90. Hmm, let me compute that.

Breaking down 650 * 90. Well, 650 * 90 is the same as 650 * 9 * 10. 650 * 9. Let me compute 650*9. 600*9 = 5400, and 50*9=450, so 5400 + 450 = 5850. Then multiplying by 10 gives 58,500. So total number of license plates is 58,500.

Wait, let me verify that multiplication again just to be sure. 650 multiplied by 90. Alternatively, 650 * 90 can be thought of as 65 * 10 * 9 * 10 = 65 * 9 * 100. 65 * 9 is 585. Then 585 * 100 is 58,500. Yep, same result. Okay, so that seems correct.

But let me just double-check all steps once again to ensure there are no mistakes.

First, letters: two different letters. Since order matters, permutations. 26P2 = 26*25 = 650. Correct.

Numbers: two different digits. Order matters, so 10P2 = 10*9 = 90. Correct.

Total combinations: 650 * 90 = 58,500. Correct.

Is there a possibility that the numbers can have leading zeros? For example, is "01" considered a valid two-digit number here? The problem says "two different one digit numbers". So each is a single digit, so the numbers are 0-9, and they just have to be different. So even if the first digit is zero, that's allowed. So for example, "01" is a valid combination. Therefore, the calculation for numbers is correct because we are considering all permutations, including those starting with zero. So no problem there.

Another thing to check: are the letters case-sensitive? The problem says "A through Z", so I assume it's uppercase only, and they are different. So no, there's no case sensitivity here. So no need to consider lowercase letters. So 26 letters total. Correct.

Are the digits allowed to be repeated? No, the problem says "two different one digit numbers", so digits must be different. Therefore, the calculation of 10*9 is correct.

So, all steps seem to check out. Therefore, the total number of license plates is 58,500.

But just to make sure, let me think of another way to approach this problem.

Alternative approach:

For the letters part: The first letter can be any of the 26 letters. The second letter can be any letter except the first one, so 25. So 26*25.

For the numbers part: Similarly, first digit can be 0-9 (10 options), second digit can be any digit except the first one, so 9. So 10*9.

Multiply them together: (26*25)*(10*9) = 650*90 = 58,500. Yep, same answer.

Alternatively, if I consider the entire license plate as a sequence of 4 characters, where the first two are letters and the last two are numbers, with all the constraints. But in that case, it's still breaking it down into two separate parts.

Another check: what if the numbers were allowed to repeat? Then it would be 10*10 = 100, but since they have to be different, it's 10*9 = 90. Similarly for letters.

Another check: if the letters were allowed to repeat, then it would be 26*26 = 676, but since they have to be different, it's 26*25 = 650.

So all these checks are consistent.

Alternatively, if I calculate the total possibilities without considering the restrictions, then subtract the ones that violate the restrictions, but that would be more complicated. Let me see:

Total number of license plates with 2 letters and 2 numbers, allowing repeats: 26*26*10*10 = 676*100 = 67,600. Then subtract the ones with repeated letters or repeated numbers. But this is inclusion-exclusion, which is more complex.

But since the problem specifies that the letters are different and the numbers are different, the first method of calculating permutations for each part is straightforward and accurate.

Alternatively, maybe someone could think that the two different letters could be arranged in some way, but since we already considered permutations, that's accounted for.

So, I think my answer is correct. Therefore, the total number of license plates is 58,500.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed, we analyze the structure: each license plate must consist of **2 different letters followed by 2 different digits**, where:

- Letters are selected from the English alphabet (A to Z).
- Digits are selected from the set of digits 0 to 9.

---

### Step 1: Calculate the number of valid letter combinations

We are choosing **2 different letters** from 26 letters (A–Z) **in order**, because the license plate has a specific sequence.

Since the letters must be **different** and **order matters**, this is a **permutation** problem.

$$
\text{Number of letter combinations} = P(26, 2) = \frac{26!}{(26 - 2)!} = 26 \times 25 = 650
$$

---

### Step 2: Calculate the number of valid digit combinations

Similarly, the two **digits must be different** and **order matters**, so again, it's a permutation of 10 digits taken 2 at a time.

$$
\text{Number of digit combinations} = P(10, 2) = \frac{10!}{(10 - 2)!} = 10 \times 9 = 90
$$

---

### Step 3: Multiply the combinations for total number of license plates

Each combination of 2 letters can be paired with any of the valid combinations of 2 digits (since the choices are independent), so we multiply the two values:

$$
\text{Total license plates} = 650 \times 90 = 58{,}500
$$

---

### Final Answer

$$
\boxed{58500}
$$

Solution: The first letter can be any of the 26 letters of the alphabet, while the second letter can be any of the 25 remaining letters. The first digit can be any of the 10 digits, while the second digit can be any of the 9 remaining digits. The number of license plates is $26\times 25\times 10\times 9=\boxed{58,500}$.


Explanation:
Okay, so I need to find the value of 'a' that makes the function f(n) continuous. Let me first recall what continuity means. If a function is continuous at a point, then the limit of the function as it approaches that point from both sides should be equal to the function's value at that point.

In this case, the function f(n) is defined piecewise, with two different expressions depending on whether n is less than 'a' or greater than or equal to 'a'. For the function to be continuous at n = a, the left-hand limit as n approaches a (from values less than a) should equal the right-hand limit as n approaches a (from values greater than or equal to a), and both should equal f(a).

But wait, since the function is defined as 4n + 3 when n < a and 7n - 12 when n ≥ a, the point where the function might have a discontinuity is at n = a. So, I need to ensure that the value of the function from the left side (approaching a from the left) is equal to the value from the right side (approaching a from the right) and also equal to f(a).

Let me write that down. For continuity at n = a, we must have:

lim(n→a⁻) f(n) = lim(n→a⁺) f(n) = f(a)

But since f(n) is defined as 4n + 3 when n < a, the left-hand limit as n approaches a would be 4a + 3. Similarly, for the right-hand limit, since n ≥ a, the function is 7n - 12, so the right-hand limit as n approaches a would be 7a - 12.

But f(a) itself is defined by the second case since n = a is included in n ≥ a, so f(a) = 7a - 12.

Therefore, for continuity, the left-hand limit must equal the right-hand limit. So:

4a + 3 = 7a - 12

Let me solve this equation for a. Subtract 4a from both sides:

3 = 3a - 12

Then, add 12 to both sides:

15 = 3a

Divide both sides by 3:

a = 5

Wait, that seems straightforward. Let me check if this makes sense. If a is 5, then for values less than 5, f(n) is 4n + 3, and for n ≥ 5, it's 7n - 12. Let me verify the continuity at n = 5.

Left-hand limit as n approaches 5: 4(5) + 3 = 20 + 3 = 23

Right-hand limit as n approaches 5: 7(5) - 12 = 35 - 12 = 23

And f(5) = 7(5) - 12 = 23. So yes, they all equal 23. Therefore, the function is continuous at n = 5.

Is there any other point where continuity might be an issue? The function is linear in both pieces, so they are continuous everywhere except possibly at the point where the definition changes, which is n = a. Therefore, by ensuring continuity at n = a, the function is continuous everywhere.

Therefore, the answer should be a = 5.

But let me think again if I made any mistake. Maybe I need to check if there's another condition? For example, sometimes in piecewise functions, especially with different domains, but here the function is defined for all real numbers? Wait, the problem didn't specify the domain of n. But since it's talking about continuity, we can assume n is a real number.

Alternatively, if n is an integer? Wait, the problem didn't specify. Hmm. Wait, the function is defined with n as the input. If n is an integer, then the function might have different considerations. Wait, but in calculus, continuity is typically discussed for real numbers. However, sometimes in discrete math, functions are defined on integers. But since the problem uses terms like "graph of y = f(n)" and "continuous", which usually implies real numbers. However, if n were an integer, then the concept of continuity might not be standard.

But the problem is from an unspecified source, but given the way it's written, I think it's safe to assume that n is a real number. So, the approach I did earlier is correct.

Wait, but if n is an integer, then for the function to be continuous, we need that the left-hand limit and right-hand limit are equal. However, if n is an integer, then approaching a from the left would mean approaching from the previous integer. But since the function is defined for all real numbers? Wait, this is confusing. The problem didn't specify.

Wait, let me check the problem statement again:

"Find a if the graph of y = f(n) is continuous."

So, the graph is continuous. If n is a real number, then the graph is continuous. If n is an integer, then the graph would be a set of discrete points, which can't be continuous. Therefore, the problem must be considering n as a real number, and the function is piecewise defined with real number inputs.

Therefore, my original approach is correct. So, solving 4a + 3 = 7a - 12 leads to a = 5.

Therefore, the answer is 5.

But just to make sure, let me test with a = 5. Let me pick a value slightly less than 5, say 4.9. Then f(4.9) = 4*(4.9) + 3 = 19.6 + 3 = 22.6.

At n = 5, f(5) = 7*5 - 12 = 35 - 12 = 23.

Now, approaching from the right, say 5.1, f(5.1) = 7*5.1 - 12 = 35.7 - 12 = 23.7.

Wait, but if we approach from the left, as n approaches 5 from the left, f(n) approaches 23. Similarly, from the right, f(n) approaches 23. So the function is continuous at n = 5.

Therefore, the value of a is indeed 5.

I think that's solid. So I don't see any mistakes in my reasoning. Therefore, the answer is 5.

**Final Answer**
The value of $ a $ is \boxed{5}.

To find the value of $ a $ such that the function

$$
f(n) =
\begin{cases}
4n + 3 & \text{if } n < a, \\
7n - 12 & \text{if } n \ge a,
\end{cases}
$$

is continuous, we need to ensure that the function is **continuous at the boundary point $ n = a $**.

---

### Understanding Continuity at $ n = a $

For a function defined piecewise to be continuous at $ n = a $, the following must hold:

$$
\lim_{n \to a^-} f(n) = \lim_{n \to a^+} f(n) = f(a)
$$

This requires the *left-hand limit* (values of $ n < a $) and the *right-hand limit* (values of $ n \ge a $) to agree, and also match the function's value at $ n = a $.

---

### Step 1: Compute the Expressions for the Limits

- For the left-hand limit as $ n \to a^- $ (i.e., using $ 4n + 3 $):
  $$
  \lim_{n \to a^-} f(n) = 4a + 3
  $$

- For the right-hand limit as $ n \to a^+ $ (i.e., using $ 7n - 12 $):
  $$
  \lim_{n \to a^+} f(n) = 7a - 12
  $$

- And since $ n = a $ is in the domain of the second case, we have:
  $$
  f(a) = 7a - 12
  $$

---

### Step 2: Equating the Left and Right Limits

For the function to be continuous at $ n = a $, we need:

$$
4a + 3 = 7a - 12
$$

Solving for $ a $:

$$
4a + 3 = 7a - 12 \\
3 + 12 = 7a - 4a \\
15 = 3a \\
a = 5
$$

---

### Step 3: Verify the Continuity at $ a = 5 $

- Left-hand limit:
  $$
  \lim_{n \to 5^-} f(n) = 4(5) + 3 = 20 + 3 = 23
  $$

- Right-hand limit:
  $$
  \lim_{n \to 5^+} f(n) = 7(5) - 12 = 35 - 12 = 23
  $$

- Function value at $ a = 5 $:
  $$
  f(5) = 7(5) - 12 = 23
  $$

All three values are equal, confirming that the function is continuous at $ n = 5 $.

---

### Final Conclusion

The only value of $ a $ that makes the function $ f(n) $ continuous is:

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of 'a' that makes the function f(n) continuous. Let me first recall what continuity means. If a function is continuous at a point, then the limit of the function as it approaches that point from both sides should be equal to the function's value at that point.

In this case, the function f(n) is defined piecewise, with two different expressions depending on whether n is less than 'a' or greater than or equal to 'a'. For the function to be continuous at n = a, the left-hand limit as n approaches a (from values less than a) should equal the right-hand limit as n approaches a (from values greater than or equal to a), and both should equal f(a).

But wait, since the function is defined as 4n + 3 when n < a and 7n - 12 when n ≥ a, the point where the function might have a discontinuity is at n = a. So, I need to ensure that the value of the function from the left side (approaching a from the left) is equal to the value from the right side (approaching a from the right) and also equal to f(a).

Let me write that down. For continuity at n = a, we must have:

lim(n→a⁻) f(n) = lim(n→a⁺) f(n) = f(a)

But since f(n) is defined as 4n + 3 when n < a, the left-hand limit as n approaches a would be 4a + 3. Similarly, for the right-hand limit, since n ≥ a, the function is 7n - 12, so the right-hand limit as n approaches a would be 7a - 12.

But f(a) itself is defined by the second case since n = a is included in n ≥ a, so f(a) = 7a - 12.

Therefore, for continuity, the left-hand limit must equal the right-hand limit. So:

4a + 3 = 7a - 12

Let me solve this equation for a. Subtract 4a from both sides:

3 = 3a - 12

Then, add 12 to both sides:

15 = 3a

Divide both sides by 3:

a = 5

Wait, that seems straightforward. Let me check if this makes sense. If a is 5, then for values less than 5, f(n) is 4n + 3, and for n ≥ 5, it's 7n - 12. Let me verify the continuity at n = 5.

Left-hand limit as n approaches 5: 4(5) + 3 = 20 + 3 = 23

Right-hand limit as n approaches 5: 7(5) - 12 = 35 - 12 = 23

And f(5) = 7(5) - 12 = 23. So yes, they all equal 23. Therefore, the function is continuous at n = 5.

Is there any other point where continuity might be an issue? The function is linear in both pieces, so they are continuous everywhere except possibly at the point where the definition changes, which is n = a. Therefore, by ensuring continuity at n = a, the function is continuous everywhere.

Therefore, the answer should be a = 5.

But let me think again if I made any mistake. Maybe I need to check if there's another condition? For example, sometimes in piecewise functions, especially with different domains, but here the function is defined for all real numbers? Wait, the problem didn't specify the domain of n. But since it's talking about continuity, we can assume n is a real number.

Alternatively, if n is an integer? Wait, the problem didn't specify. Hmm. Wait, the function is defined with n as the input. If n is an integer, then the function might have different considerations. Wait, but in calculus, continuity is typically discussed for real numbers. However, sometimes in discrete math, functions are defined on integers. But since the problem uses terms like "graph of y = f(n)" and "continuous", which usually implies real numbers. However, if n were an integer, then the concept of continuity might not be standard.

But the problem is from an unspecified source, but given the way it's written, I think it's safe to assume that n is a real number. So, the approach I did earlier is correct.

Wait, but if n is an integer, then for the function to be continuous, we need that the left-hand limit and right-hand limit are equal. However, if n is an integer, then approaching a from the left would mean approaching from the previous integer. But since the function is defined for all real numbers? Wait, this is confusing. The problem didn't specify.

Wait, let me check the problem statement again:

"Find a if the graph of y = f(n) is continuous."

So, the graph is continuous. If n is a real number, then the graph is continuous. If n is an integer, then the graph would be a set of discrete points, which can't be continuous. Therefore, the problem must be considering n as a real number, and the function is piecewise defined with real number inputs.

Therefore, my original approach is correct. So, solving 4a + 3 = 7a - 12 leads to a = 5.

Therefore, the answer is 5.

But just to make sure, let me test with a = 5. Let me pick a value slightly less than 5, say 4.9. Then f(4.9) = 4*(4.9) + 3 = 19.6 + 3 = 22.6.

At n = 5, f(5) = 7*5 - 12 = 35 - 12 = 23.

Now, approaching from the right, say 5.1, f(5.1) = 7*5.1 - 12 = 35.7 - 12 = 23.7.

Wait, but if we approach from the left, as n approaches 5 from the left, f(n) approaches 23. Similarly, from the right, f(n) approaches 23. So the function is continuous at n = 5.

Therefore, the value of a is indeed 5.

I think that's solid. So I don't see any mistakes in my reasoning. Therefore, the answer is 5.

**Final Answer**
The value of $ a $ is \boxed{5}.

To find the value of $ a $ such that the function

$$
f(n) =
\begin{cases}
4n + 3 & \text{if } n < a, \\
7n - 12 & \text{if } n \ge a,
\end{cases}
$$

is continuous, we need to ensure that the function is **continuous at the boundary point $ n = a $**.

---

### Understanding Continuity at $ n = a $

For a function defined piecewise to be continuous at $ n = a $, the following must hold:

$$
\lim_{n \to a^-} f(n) = \lim_{n \to a^+} f(n) = f(a)
$$

This requires the *left-hand limit* (values of $ n < a $) and the *right-hand limit* (values of $ n \ge a $) to agree, and also match the function's value at $ n = a $.

---

### Step 1: Compute the Expressions for the Limits

- For the left-hand limit as $ n \to a^- $ (i.e., using $ 4n + 3 $):
  $$
  \lim_{n \to a^-} f(n) = 4a + 3
  $$

- For the right-hand limit as $ n \to a^+ $ (i.e., using $ 7n - 12 $):
  $$
  \lim_{n \to a^+} f(n) = 7a - 12
  $$

- And since $ n = a $ is in the domain of the second case, we have:
  $$
  f(a) = 7a - 12
  $$

---

### Step 2: Equating the Left and Right Limits

For the function to be continuous at $ n = a $, we need:

$$
4a + 3 = 7a - 12
$$

Solving for $ a $:

$$
4a + 3 = 7a - 12 \\
3 + 12 = 7a - 4a \\
15 = 3a \\
a = 5
$$

---

### Step 3: Verify the Continuity at $ a = 5 $

- Left-hand limit:
  $$
  \lim_{n \to 5^-} f(n) = 4(5) + 3 = 20 + 3 = 23
  $$

- Right-hand limit:
  $$
  \lim_{n \to 5^+} f(n) = 7(5) - 12 = 35 - 12 = 23
  $$

- Function value at $ a = 5 $:
  $$
  f(5) = 7(5) - 12 = 23
  $$

All three values are equal, confirming that the function is continuous at $ n = 5 $.

---

### Final Conclusion

The only value of $ a $ that makes the function $ f(n) $ continuous is:

$$
\boxed{5}
$$

Solution: In order for the graph to be continuous, the two pieces of the function must meet at $n=a$. In order for this to happen, we know that $4a+3=7a-12$. Solving for $a$, we find that $a=\frac{15}{3}=\boxed{5}$.