Explanation:
Okay, so I need to figure out how many integers satisfy the inequality |x - 2| ≤ 5.6. Hmm, let me start by recalling what absolute value inequalities mean. The expression |x - 2| ≤ 5.6 means that the distance between x and 2 is less than or equal to 5.6. So, I can rewrite this inequality without the absolute value by considering the definition of absolute value.

Right, if |a| ≤ b, then -b ≤ a ≤ b. Applying that here, I can write:

-5.6 ≤ x - 2 ≤ 5.6

Now, I need to solve for x. To do that, I should add 2 to all three parts of the inequality. Let me do that step by step.

Starting with the left part: -5.6 + 2. Let me calculate that. Hmm, -5.6 + 2 is the same as -(5.6 - 2) which is -3.6? Wait, no. Wait, adding 2 to -5.6. Let me think. If I have -5.6 and add 2, that's like moving 2 units to the right on the number line. So, -5.6 + 2 = -3.6. Yeah, that's right.

Then the middle part is x - 2 + 2, which simplifies to x.

And the right part is 5.6 + 2, which is 7.6.

So putting it all together, the inequality becomes:

-3.6 ≤ x ≤ 7.6

So, x has to be greater than or equal to -3.6 and less than or equal to 7.6. Now, the question is asking for the number of integers in this solution set.

Alright, so I need to find all integers x such that -3.6 ≤ x ≤ 7.6. Let me think about how to approach this.

First, let me note that integers are whole numbers, so they don't include fractions or decimals. So, I need to find all integers between -3.6 and 7.6, including the endpoints if they are integers.

But wait, the endpoints here are -3.6 and 7.6. Are these integers? No, they are not. So, the integers that satisfy the inequality will be the integers greater than or equal to the next integer greater than -3.6 and less than or equal to the next integer less than 7.6.

Let me clarify. Since x has to be greater than or equal to -3.6, the smallest integer that satisfies this is -3, because -3 is greater than -3.6. Similarly, since x has to be less than or equal to 7.6, the largest integer that satisfies this is 7, because 7 is less than 7.6.

Therefore, the integers in the solution set are from -3 up to 7, inclusive. Now, I need to count how many integers are in this range.

To count the integers from -3 to 7 inclusive, I can use the formula: number of integers = (last - first) + 1.

Let me check that formula. For example, from 1 to 3 inclusive, that's 3 - 1 + 1 = 3 numbers, which is correct. So applying that here:

Last integer is 7, first integer is -3. So, 7 - (-3) + 1 = 7 + 3 + 1 = 11. Wait, 7 - (-3) is 10, then +1 is 11. So, 11 integers? Let me verify by listing them.

Starting from -3: -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. Let me count them:

1. -3
2. -2
3. -1
4. 0
5. 1
6. 2
7. 3
8. 4
9. 5
10. 6
11. 7

Yes, that's 11 integers. So, the answer should be 11.

But wait, let me make sure I didn't make a mistake in determining the range. Let me check the original inequality again.

Original inequality: |x - 2| ≤ 5.6. So solving that gives -5.6 ≤ x - 2 ≤ 5.6, then adding 2 gives -3.6 ≤ x ≤ 7.6. So, x is between -3.6 and 7.6.

So, integers greater than or equal to -3.6. The integers greater than or equal to -3.6 are -3, -2, -1, 0, 1, 2, 3,... up to 7.6. So, the upper bound is 7.6, so integers less than or equal to 7.6 are up to 7.

But let me check if maybe there's an integer that's just below 7.6, which is 7, and above -3.6, which is -3. So, yes, the integers from -3 to 7 inclusive.

But let me check if maybe there's a mistake here. For example, if the upper bound was 7.6, does that include 7.6? Well, the inequality is x ≤ 7.6, so 7.6 is not an integer, so the largest integer less than or equal to 7.6 is 7. Similarly, the smallest integer greater than or equal to -3.6 is -3.

But let me check if maybe there's a case where the endpoints are integers. For example, if the inequality was |x - 2| ≤ 5, then the solution would be -3 ≤ x ≤ 7, and the integers would be from -3 to 7 inclusive, which is 11 numbers as well. Wait, but in this case, the original inequality is |x - 2| ≤ 5.6, which is a bit more than 5. So, the interval is a bit wider.

But since 5.6 is not an integer, the endpoints of the interval are not integers, so the integers in the interval are still from -3 to 7.

But let me check if maybe there's an integer that's just below -3.6? For example, -4. Is -4 greater than or equal to -3.6? No, because -4 is less than -3.6. So, -4 is not included. Similarly, 8 is greater than 7.6, so 8 is not included.

Therefore, the integers are indeed from -3 to 7 inclusive.

But let me count again. From -3 to 7 inclusive. How many numbers is that?

One way to calculate is to take the number of numbers from 0 to 7 inclusive, which is 8 numbers (0,1,2,3,4,5,6,7), then add the numbers from -3 to -1 inclusive, which is 3 numbers (-3,-2,-1). So total is 8 + 3 = 11.

Alternatively, using the formula: (last - first) + 1. So, 7 - (-3) + 1 = 7 + 3 + 1 = 11.

So, that seems correct.

But let me think again. Maybe I can visualize the number line. Starting at -3.6, so between -4 and -3. The first integer is -3. Then going up to 7.6, so between 7 and 8. The last integer is 7. So, the integers from -3 to 7.

Alternatively, maybe I can think of the number of integers between -3.6 and 7.6. Since the interval is from -3.6 to 7.6, which is 11.2 units long. But the number of integers isn't directly related to the length, but rather the positions.

Alternatively, maybe I can think of the integers that satisfy the inequality. Let me list them again:

Starting from -3:

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.

Yes, that's 11 numbers.

Wait, but let me check if maybe -3.6 is included. Since the inequality is |x - 2| ≤ 5.6, so x can be as low as -3.6. But since -3.6 is not an integer, the integers that are greater than or equal to -3.6 are all integers greater than -3.6, which is -3 and above. Similarly, integers less than or equal to 7.6 are 7 and below.

So, yes, the integers from -3 to 7 inclusive.

But let me check if maybe there's a mistake in solving the inequality. Let me re-solve it step by step.

Original inequality: |x - 2| ≤ 5.6.

This is equivalent to -5.6 ≤ x - 2 ≤ 5.6.

Adding 2 to all parts:

Left side: -5.6 + 2 = -3.6

Middle: x - 2 + 2 = x

Right side: 5.6 + 2 = 7.6

So, the inequality is -3.6 ≤ x ≤ 7.6.

Yes, that's correct.

So, the solution set is all real numbers between -3.6 and 7.6, inclusive.

Therefore, the integers in this interval are the integers from -3 up to 7.

So, the count is 11.

But let me check once more by considering the endpoints.

For example, if x = -3, then | -3 - 2 | = | -5 | = 5 ≤ 5.6. So, that's okay.

If x = 7, then |7 - 2| = |5| = 5 ≤ 5.6. That's also okay.

What about x = -4? | -4 - 2 | = | -6 | = 6, which is greater than 5.6. So, not included.

What about x = 8? |8 - 2| = 6, which is also greater than 5.6. Not included.

So, yes, -3 and 7 are included, but -4 and 8 are not.

Therefore, the integers are from -3 to 7 inclusive.

So, the number of integers is 11.

But just to make sure, let me count them again:

Starting from -3:

1. -3

2. -2

3. -1

4. 0

5. 1

6. 2

7. 3

8. 4

9. 5

10. 6

11. 7

Yep, that's 11 integers.

Therefore, the answer is 11.

But wait, just to make sure there isn't a trick here. Sometimes with absolute value inequalities, there can be edge cases, but in this case, since the numbers are not integers, and the interval is straightforward, I think this is correct.

Alternatively, maybe I can think of the number of integers in the interval [a, b] where a and b are real numbers. The formula is floor(b) - ceil(a) + 1.

So, in this case, a is -3.6, so ceil(a) is -3.

b is 7.6, so floor(b) is 7.

Therefore, number of integers is 7 - (-3) + 1 = 7 + 3 + 1 = 11.

Yes, that's the same result.

Therefore, I can be confident that the number of integers in the solution set is 11.

**Final Answer**
The number of integers in the solution set is \boxed{11}.

To determine how many integers satisfy the inequality $ |x - 2| \leq 5.6 $, we begin by interpreting the absolute value inequality.

---

### Step 1: Rewrite the Inequality

The inequality $ |x - 2| \leq 5.6 $ means that the distance between $ x $ and 2 is at most 5.6. This can be rewritten as a compound inequality:

$$
-5.6 \leq x - 2 \leq 5.6
$$

---

### Step 2: Solve for $ x $

To isolate $ x $, add 2 to all parts of the inequality:

$$
-5.6 + 2 \leq x \leq 5.6 + 2
$$

$$
-3.6 \leq x \leq 7.6
$$

So, the solution set is all real numbers between $ -3.6 $ and $ 7.6 $, inclusive.

---

### Step 3: Identify the Integer Solutions

We are interested in the **integers** within this interval. Since $ -3.6 $ and $ 7.6 $ are not integers, we find the smallest integer greater than or equal to $ -3.6 $, which is $ -3 $, and the largest integer less than or equal to $ 7.6 $, which is $ 7 $.

Thus, the integers in the solution set are:

$$
-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7
$$

---

### Step 4: Count the Integers

We can count the number of integers from $ -3 $ to $ 7 $, inclusive. Using the formula for the number of integers in a range:

$$
\text{Number of integers} = \text{last} - \text{first} + 1
$$

$$
= 7 - (-3) + 1 = 7 + 3 + 1 = 11
$$

---

### Final Answer

$$
\boxed{11}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many integers satisfy the inequality |x - 2| ≤ 5.6. Hmm, let me start by recalling what absolute value inequalities mean. The expression |x - 2| ≤ 5.6 means that the distance between x and 2 is less than or equal to 5.6. So, I can rewrite this inequality without the absolute value by considering the definition of absolute value.

Right, if |a| ≤ b, then -b ≤ a ≤ b. Applying that here, I can write:

-5.6 ≤ x - 2 ≤ 5.6

Now, I need to solve for x. To do that, I should add 2 to all three parts of the inequality. Let me do that step by step.

Starting with the left part: -5.6 + 2. Let me calculate that. Hmm, -5.6 + 2 is the same as -(5.6 - 2) which is -3.6? Wait, no. Wait, adding 2 to -5.6. Let me think. If I have -5.6 and add 2, that's like moving 2 units to the right on the number line. So, -5.6 + 2 = -3.6. Yeah, that's right.

Then the middle part is x - 2 + 2, which simplifies to x.

And the right part is 5.6 + 2, which is 7.6.

So putting it all together, the inequality becomes:

-3.6 ≤ x ≤ 7.6

So, x has to be greater than or equal to -3.6 and less than or equal to 7.6. Now, the question is asking for the number of integers in this solution set.

Alright, so I need to find all integers x such that -3.6 ≤ x ≤ 7.6. Let me think about how to approach this.

First, let me note that integers are whole numbers, so they don't include fractions or decimals. So, I need to find all integers between -3.6 and 7.6, including the endpoints if they are integers.

But wait, the endpoints here are -3.6 and 7.6. Are these integers? No, they are not. So, the integers that satisfy the inequality will be the integers greater than or equal to the next integer greater than -3.6 and less than or equal to the next integer less than 7.6.

Let me clarify. Since x has to be greater than or equal to -3.6, the smallest integer that satisfies this is -3, because -3 is greater than -3.6. Similarly, since x has to be less than or equal to 7.6, the largest integer that satisfies this is 7, because 7 is less than 7.6.

Therefore, the integers in the solution set are from -3 up to 7, inclusive. Now, I need to count how many integers are in this range.

To count the integers from -3 to 7 inclusive, I can use the formula: number of integers = (last - first) + 1.

Let me check that formula. For example, from 1 to 3 inclusive, that's 3 - 1 + 1 = 3 numbers, which is correct. So applying that here:

Last integer is 7, first integer is -3. So, 7 - (-3) + 1 = 7 + 3 + 1 = 11. Wait, 7 - (-3) is 10, then +1 is 11. So, 11 integers? Let me verify by listing them.

Starting from -3: -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. Let me count them:

1. -3
2. -2
3. -1
4. 0
5. 1
6. 2
7. 3
8. 4
9. 5
10. 6
11. 7

Yes, that's 11 integers. So, the answer should be 11.

But wait, let me make sure I didn't make a mistake in determining the range. Let me check the original inequality again.

Original inequality: |x - 2| ≤ 5.6. So solving that gives -5.6 ≤ x - 2 ≤ 5.6, then adding 2 gives -3.6 ≤ x ≤ 7.6. So, x is between -3.6 and 7.6.

So, integers greater than or equal to -3.6. The integers greater than or equal to -3.6 are -3, -2, -1, 0, 1, 2, 3,... up to 7.6. So, the upper bound is 7.6, so integers less than or equal to 7.6 are up to 7.

But let me check if maybe there's an integer that's just below 7.6, which is 7, and above -3.6, which is -3. So, yes, the integers from -3 to 7 inclusive.

But let me check if maybe there's a mistake here. For example, if the upper bound was 7.6, does that include 7.6? Well, the inequality is x ≤ 7.6, so 7.6 is not an integer, so the largest integer less than or equal to 7.6 is 7. Similarly, the smallest integer greater than or equal to -3.6 is -3.

But let me check if maybe there's a case where the endpoints are integers. For example, if the inequality was |x - 2| ≤ 5, then the solution would be -3 ≤ x ≤ 7, and the integers would be from -3 to 7 inclusive, which is 11 numbers as well. Wait, but in this case, the original inequality is |x - 2| ≤ 5.6, which is a bit more than 5. So, the interval is a bit wider.

But since 5.6 is not an integer, the endpoints of the interval are not integers, so the integers in the interval are still from -3 to 7.

But let me check if maybe there's an integer that's just below -3.6? For example, -4. Is -4 greater than or equal to -3.6? No, because -4 is less than -3.6. So, -4 is not included. Similarly, 8 is greater than 7.6, so 8 is not included.

Therefore, the integers are indeed from -3 to 7 inclusive.

But let me count again. From -3 to 7 inclusive. How many numbers is that?

One way to calculate is to take the number of numbers from 0 to 7 inclusive, which is 8 numbers (0,1,2,3,4,5,6,7), then add the numbers from -3 to -1 inclusive, which is 3 numbers (-3,-2,-1). So total is 8 + 3 = 11.

Alternatively, using the formula: (last - first) + 1. So, 7 - (-3) + 1 = 7 + 3 + 1 = 11.

So, that seems correct.

But let me think again. Maybe I can visualize the number line. Starting at -3.6, so between -4 and -3. The first integer is -3. Then going up to 7.6, so between 7 and 8. The last integer is 7. So, the integers from -3 to 7.

Alternatively, maybe I can think of the number of integers between -3.6 and 7.6. Since the interval is from -3.6 to 7.6, which is 11.2 units long. But the number of integers isn't directly related to the length, but rather the positions.

Alternatively, maybe I can think of the integers that satisfy the inequality. Let me list them again:

Starting from -3:

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.

Yes, that's 11 numbers.

Wait, but let me check if maybe -3.6 is included. Since the inequality is |x - 2| ≤ 5.6, so x can be as low as -3.6. But since -3.6 is not an integer, the integers that are greater than or equal to -3.6 are all integers greater than -3.6, which is -3 and above. Similarly, integers less than or equal to 7.6 are 7 and below.

So, yes, the integers from -3 to 7 inclusive.

But let me check if maybe there's a mistake in solving the inequality. Let me re-solve it step by step.

Original inequality: |x - 2| ≤ 5.6.

This is equivalent to -5.6 ≤ x - 2 ≤ 5.6.

Adding 2 to all parts:

Left side: -5.6 + 2 = -3.6

Middle: x - 2 + 2 = x

Right side: 5.6 + 2 = 7.6

So, the inequality is -3.6 ≤ x ≤ 7.6.

Yes, that's correct.

So, the solution set is all real numbers between -3.6 and 7.6, inclusive.

Therefore, the integers in this interval are the integers from -3 up to 7.

So, the count is 11.

But let me check once more by considering the endpoints.

For example, if x = -3, then | -3 - 2 | = | -5 | = 5 ≤ 5.6. So, that's okay.

If x = 7, then |7 - 2| = |5| = 5 ≤ 5.6. That's also okay.

What about x = -4? | -4 - 2 | = | -6 | = 6, which is greater than 5.6. So, not included.

What about x = 8? |8 - 2| = 6, which is also greater than 5.6. Not included.

So, yes, -3 and 7 are included, but -4 and 8 are not.

Therefore, the integers are from -3 to 7 inclusive.

So, the number of integers is 11.

But just to make sure, let me count them again:

Starting from -3:

1. -3

2. -2

3. -1

4. 0

5. 1

6. 2

7. 3

8. 4

9. 5

10. 6

11. 7

Yep, that's 11 integers.

Therefore, the answer is 11.

But wait, just to make sure there isn't a trick here. Sometimes with absolute value inequalities, there can be edge cases, but in this case, since the numbers are not integers, and the interval is straightforward, I think this is correct.

Alternatively, maybe I can think of the number of integers in the interval [a, b] where a and b are real numbers. The formula is floor(b) - ceil(a) + 1.

So, in this case, a is -3.6, so ceil(a) is -3.

b is 7.6, so floor(b) is 7.

Therefore, number of integers is 7 - (-3) + 1 = 7 + 3 + 1 = 11.

Yes, that's the same result.

Therefore, I can be confident that the number of integers in the solution set is 11.

**Final Answer**
The number of integers in the solution set is \boxed{11}.

To determine how many integers satisfy the inequality $ |x - 2| \leq 5.6 $, we begin by interpreting the absolute value inequality.

---

### Step 1: Rewrite the Inequality

The inequality $ |x - 2| \leq 5.6 $ means that the distance between $ x $ and 2 is at most 5.6. This can be rewritten as a compound inequality:

$$
-5.6 \leq x - 2 \leq 5.6
$$

---

### Step 2: Solve for $ x $

To isolate $ x $, add 2 to all parts of the inequality:

$$
-5.6 + 2 \leq x \leq 5.6 + 2
$$

$$
-3.6 \leq x \leq 7.6
$$

So, the solution set is all real numbers between $ -3.6 $ and $ 7.6 $, inclusive.

---

### Step 3: Identify the Integer Solutions

We are interested in the **integers** within this interval. Since $ -3.6 $ and $ 7.6 $ are not integers, we find the smallest integer greater than or equal to $ -3.6 $, which is $ -3 $, and the largest integer less than or equal to $ 7.6 $, which is $ 7 $.

Thus, the integers in the solution set are:

$$
-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7
$$

---

### Step 4: Count the Integers

We can count the number of integers from $ -3 $ to $ 7 $, inclusive. Using the formula for the number of integers in a range:

$$
\text{Number of integers} = \text{last} - \text{first} + 1
$$

$$
= 7 - (-3) + 1 = 7 + 3 + 1 = 11
$$

---

### Final Answer

$$
\boxed{11}
$$

Solution: Getting rid of the absolute value, we have $-5.6 \le x-2 \le 5.6$, or $-3.6 \le x \le 7.6$. Thus, $x$ can be any integer from -3 to 7, inclusive. There are $7-(-3)+1=\boxed{11}$ integers in this range.


Explanation:
Okay, so I need to find sin T in triangle RST. They told me that sin R is 2/5. Let me try to visualize the triangle first. The Asymptote code shows that T is at (0,0), S is at (2,0), and R is at (2, sqrt(21)). There's a right angle mark at R, S, T. Wait, hold on, right angle mark is usually at the vertex where the right angle is. The Asymptote code says rightanglemark(R,S,T,10), so that means the right angle is at S? Because the right angle mark is between R, S, T. So the right angle is at S? Let me confirm.

In the Asymptote code, they draw R connected to S connected to T connected to R. Then the right angle mark is at R, S, T. Wait, the rightanglemark is usually drawn between three points, with the right angle at the middle one. So if it's rightanglemark(R,S,T,10), then the right angle is at S. So triangle RST is right-angled at S. Therefore, angle S is 90 degrees. So the triangle has a right angle at S, with legs RS and ST, and hypotenuse RT.

Wait, but the Asymptote code labels the side RT as 5. Let me check. The Asymptote code says label("$5$",(R+T)/2,NW);. The midpoint between R and T is ( (2 + 0)/2, (sqrt(21) + 0)/2 ) which is (1, sqrt(21)/2). So they label the segment RT as 5. Therefore, RT is the hypotenuse of the right triangle, and its length is 5. So triangle RST is right-angled at S, with hypotenuse RT = 5. Then RS and ST are the legs.

But they also mention sin R = 2/5. Let me recall that in a right-angled triangle, the sine of an angle is opposite over hypotenuse. But angle R is at vertex R. Wait, if the triangle is right-angled at S, then angles at R and T are acute angles. So angle R is at point R, which is connected to S and T. Let me try to sketch mentally.

Point S is at (2,0), T is at (0,0), R is at (2, sqrt(21)). So RS is vertical from (2,0) to (2, sqrt(21)), and ST is horizontal from (2,0) to (0,0). So the right angle is at S, between RS and ST. Therefore, angle at S is 90 degrees. Then angle R is at point R, between RS and RT. Similarly, angle T is at point T, between ST and RT.

So, given that, sin R is 2/5. Let me recall that in triangle RST, angle R is opposite to side ST. Wait, in a triangle, the sine of an angle is opposite over hypotenuse. Wait, but since it's a right-angled triangle, hypotenuse is RT. So for angle R, the opposite side is ST, and the hypotenuse is RT. Therefore, sin R = opposite / hypotenuse = ST / RT = 2/5. But they told us that RT is 5, so ST / 5 = 2/5, so ST = 2. Therefore, ST is 2 units long. Then, since the triangle is right-angled at S, we can find RS using Pythagoras. RT^2 = RS^2 + ST^2. So 5^2 = RS^2 + 2^2 => 25 = RS^2 + 4 => RS^2 = 21 => RS = sqrt(21). Which matches the coordinates given in the Asymptote code, since R is at (2, sqrt(21)), so RS is vertical from (2,0) to (2, sqrt(21)), so length sqrt(21). So that checks out.

Now, the question is asking for sin T. So angle T is at point T, which is the right angle? Wait no, angle T is at point T, which is connected to S and R. Since the right angle is at S, angle T is one of the acute angles. So in triangle RST, angle T is opposite to side RS. Wait, let me think again. In triangle RST, angle T is at point T. The sides opposite to angles are as follows: side opposite angle R is ST, side opposite angle S is RT, and side opposite angle T is RS. Therefore, sin T would be opposite over hypotenuse. The hypotenuse is RT, which is 5. The side opposite angle T is RS, which is sqrt(21). Therefore, sin T = RS / RT = sqrt(21)/5. But let me check if that's correct.

Alternatively, maybe I can use the coordinates to compute it. Since coordinates are given, maybe I can compute the lengths and then compute the sine. Let me confirm.

Coordinates:

T is at (0,0), S is at (2,0), R is at (2, sqrt(21)).

So, angle T is at (0,0). Let me find the sides adjacent and opposite to angle T. In triangle RST, angle T is between sides ST and RT. So, to find sin T, we can think of the opposite side over hypotenuse. Wait, but in a right-angled triangle, the hypotenuse is opposite the right angle. Since the right angle is at S, hypotenuse is RT. Therefore, for angle T, which is at (0,0), the sides are:

- Opposite side: RS, which is sqrt(21)

- Adjacent side: ST, which is 2

- Hypotenuse: RT, which is 5

Therefore, sin T = opposite / hypotenuse = RS / RT = sqrt(21)/5. Alternatively, if I think of angle T, then in terms of the triangle, sin T = opposite / hypotenuse. So that would be RS / RT. Which is sqrt(21)/5. So that would be the answer. But let me check if that's correct.

Alternatively, maybe I can compute it using coordinates. Let me compute the coordinates. The coordinates of T are (0,0), S is (2,0), R is (2, sqrt(21)). So vector from T to S is (2,0), and vector from T to R is (2, sqrt(21)). So angle at T is between vectors TS and TR. To find sin T, maybe I can compute the angle between those two vectors.

Alternatively, since we know the lengths of the sides, we can use trigonometry. Since we have a right-angled triangle, we can use the sides. Since angle T is adjacent to side ST (length 2) and opposite to side RS (length sqrt(21)), so sin T = opposite / hypotenuse = sqrt(21)/5. Therefore, the answer is sqrt(21)/5. But let me check if that's the case.

Alternatively, maybe I can use the fact that in a right-angled triangle, the two non-right angles are complementary. So angle R + angle T = 90 degrees. Therefore, sin T = sin(90 - R) = cos R. So if I can find cos R, then that would be sin T. Since sin R is 2/5, then cos R can be found using sin^2 R + cos^2 R = 1. Therefore, cos R = sqrt(1 - (2/5)^2) = sqrt(1 - 4/25) = sqrt(21/25) = sqrt(21)/5. Therefore, sin T = cos R = sqrt(21)/5. So that's another way to get the same answer. Therefore, sin T is sqrt(21)/5.

But let me check if that's consistent with the coordinates. Let me compute the coordinates. The coordinates of the triangle are T(0,0), S(2,0), R(2, sqrt(21)). So angle at T. Let me compute the lengths of the sides adjacent to angle T. The sides adjacent to angle T are ST and RT? Wait, no. In angle T, the sides are ST and RT. Wait, but in a triangle, the sides adjacent to angle T are the two sides that form the angle, which are ST and RT. However, in the right-angled triangle, the hypotenuse is RT, and the other sides are ST and RS. Wait, maybe I need to think again.

Wait, angle T is at point T. The sides forming angle T are ST and RT. The side opposite angle T is RS. So, in terms of trigonometry, sin T = opposite / hypotenuse = RS / RT = sqrt(21)/5. Which is the same as before. Alternatively, if I use coordinates, I can compute the angle. Let me compute the coordinates.

Vector TS is from T to S: (2,0) - (0,0) = (2,0). Vector TR is from T to R: (2, sqrt(21)) - (0,0) = (2, sqrt(21)). The angle between vectors TS and TR is angle T. To find sin T, we can use the cross product formula. The magnitude of the cross product of vectors TS and TR is |TS x TR| = |(2)(sqrt(21)) - (0)(2)| = 2*sqrt(21). The magnitude of TS is sqrt(2^2 + 0^2) = 2. The magnitude of TR is sqrt(2^2 + (sqrt(21))^2) = sqrt(4 + 21) = sqrt(25) = 5. Then, the sine of the angle between them is |TS x TR| / (|TS| |TR|) = 2*sqrt(21) / (2*5) = sqrt(21)/5. Which matches the previous results. Therefore, sin T is sqrt(21)/5. Therefore, the answer is sqrt(21)/5.

But let me check again if I didn't make any mistake. The problem says sin R is 2/5. Let me check that with coordinates. In triangle RST, angle R is at point R. The sides adjacent to angle R are RS and RT. Wait, no. In angle R, the sides are RS and RT. Wait, but in the triangle, angle R is between RS and RT. Wait, but since the triangle is right-angled at S, angle R is between RS and RT. Wait, but RS is vertical from S to R, and RT is the hypotenuse from R to T. So, to compute sin R, which is opposite over hypotenuse. The side opposite angle R is ST, which is 2, and hypotenuse is RT, which is 5. Therefore, sin R = 2/5, which matches the given information. Therefore, that's correct. Therefore, sin T is sqrt(21)/5. Therefore, the answer is sqrt(21)/5. So I think that's correct.

Alternatively, maybe I can use the area of the triangle. The area is (1/2)*ST*RS = (1/2)*2*sqrt(21) = sqrt(21). Also, area can be expressed as (1/2)*RT*height from S. Wait, but maybe not necessary. Alternatively, since we know all sides, maybe not needed.

Alternatively, since we have coordinates, we can compute the angle. For example, the coordinates of T(0,0), S(2,0), R(2, sqrt(21)). So, angle at T. The coordinates can be used to compute the slopes of the sides, but maybe that's more complicated. Alternatively, using trigonometry. Since we have the lengths, we can compute tan T. In triangle RST, tan T = opposite / adjacent = RS / ST = sqrt(21)/2. Therefore, sin T = tan T / sqrt(1 + tan^2 T) = [sqrt(21)/2] / sqrt(1 + (21/4)) = [sqrt(21)/2] / sqrt(25/4) = [sqrt(21)/2] / [5/2] = sqrt(21)/5. Which again gives the same result. Therefore, sin T is sqrt(21)/5.

Therefore, after multiple methods, I get the same answer. Therefore, I think that is correct. Therefore, the answer is sqrt(21)/5. So in boxed form, that would be \boxed{\dfrac{\sqrt{21}}{5}}.

**Final Answer**
\boxed{\dfrac{\sqrt{21}}{5}}

To determine $\sin T$ in triangle $RST$, we are given that $\sin R = \frac{2}{5}$ and that the triangle is right-angled at $S$. Let's proceed step by step using both geometric and trigonometric reasoning.

---

### Step 1: Understand the Triangle

From the Asymptote code and the description:

- $T = (0, 0)$
- $S = (2, 0)$
- $R = (2, \sqrt{21})$

This places the triangle in the coordinate plane with:

- $S$ at the origin,
- $T$ to the right of $S$,
- $R$ directly above $S$.

The right angle is at $S$, so triangle $RST$ is a **right triangle** with the right angle at $S$. Therefore:

- $RT$ is the **hypotenuse**,
- $RS$ and $ST$ are the **legs**.

---

### Step 2: Use Given Information

We are told that $\sin R = \frac{2}{5}$. In a right triangle, the sine of an angle is the ratio of the **opposite side** to the **hypotenuse**.

- Angle $R$ is at point $R$, so the side opposite to angle $R$ is $ST$.
- The hypotenuse is $RT$.

So:

$$
\sin R = \frac{ST}{RT} = \frac{2}{5}
$$

From this, we can deduce that:

$$
RT = 5 \quad \text{and} \quad ST = 2
$$

---

### Step 3: Use the Pythagorean Theorem to Find $RS$

Since triangle $RST$ is right-angled at $S$, we can apply the Pythagorean Theorem:

$$
RT^2 = RS^2 + ST^2
$$

$$
5^2 = RS^2 + 2^2 \Rightarrow 25 = RS^2 + 4 \Rightarrow RS^2 = 21 \Rightarrow RS = \sqrt{21}
$$

---

### Step 4: Find $\sin T$

Angle $T$ is at point $T$, and in triangle $RST$, the side opposite to angle $T$ is $RS$, and the hypotenuse is $RT$. So:

$$
\sin T = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{RS}{RT} = \frac{\sqrt{21}}{5}
$$

---

### Step 5: Confirm Using Complementary Angles

In a right triangle, the two non-right angles are **complementary**. That is:

$$
\angle R + \angle T = 90^\circ
$$

So:

$$
\sin T = \cos R
$$

We are given $\sin R = \frac{2}{5}$, so:

$$
\cos R = \sqrt{1 - \sin^2 R} = \sqrt{1 - \left(\frac{2}{5}\right)^2} = \sqrt{1 - \frac{4}{25}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}
$$

Thus, $\sin T = \cos R = \frac{\sqrt{21}}{5}$, confirming our earlier result.

---

### Final Answer

$$
\boxed{\dfrac{\sqrt{21}}{5}}
$$

[Assistant]
Explanation:
Okay, so I need to find sin T in triangle RST. They told me that sin R is 2/5. Let me try to visualize the triangle first. The Asymptote code shows that T is at (0,0), S is at (2,0), and R is at (2, sqrt(21)). There's a right angle mark at R, S, T. Wait, hold on, right angle mark is usually at the vertex where the right angle is. The Asymptote code says rightanglemark(R,S,T,10), so that means the right angle is at S? Because the right angle mark is between R, S, T. So the right angle is at S? Let me confirm.

In the Asymptote code, they draw R connected to S connected to T connected to R. Then the right angle mark is at R, S, T. Wait, the rightanglemark is usually drawn between three points, with the right angle at the middle one. So if it's rightanglemark(R,S,T,10), then the right angle is at S. So triangle RST is right-angled at S. Therefore, angle S is 90 degrees. So the triangle has a right angle at S, with legs RS and ST, and hypotenuse RT.

Wait, but the Asymptote code labels the side RT as 5. Let me check. The Asymptote code says label("$5$",(R+T)/2,NW);. The midpoint between R and T is ( (2 + 0)/2, (sqrt(21) + 0)/2 ) which is (1, sqrt(21)/2). So they label the segment RT as 5. Therefore, RT is the hypotenuse of the right triangle, and its length is 5. So triangle RST is right-angled at S, with hypotenuse RT = 5. Then RS and ST are the legs.

But they also mention sin R = 2/5. Let me recall that in a right-angled triangle, the sine of an angle is opposite over hypotenuse. But angle R is at vertex R. Wait, if the triangle is right-angled at S, then angles at R and T are acute angles. So angle R is at point R, which is connected to S and T. Let me try to sketch mentally.

Point S is at (2,0), T is at (0,0), R is at (2, sqrt(21)). So RS is vertical from (2,0) to (2, sqrt(21)), and ST is horizontal from (2,0) to (0,0). So the right angle is at S, between RS and ST. Therefore, angle at S is 90 degrees. Then angle R is at point R, between RS and RT. Similarly, angle T is at point T, between ST and RT.

So, given that, sin R is 2/5. Let me recall that in triangle RST, angle R is opposite to side ST. Wait, in a triangle, the sine of an angle is opposite over hypotenuse. Wait, but since it's a right-angled triangle, hypotenuse is RT. So for angle R, the opposite side is ST, and the hypotenuse is RT. Therefore, sin R = opposite / hypotenuse = ST / RT = 2/5. But they told us that RT is 5, so ST / 5 = 2/5, so ST = 2. Therefore, ST is 2 units long. Then, since the triangle is right-angled at S, we can find RS using Pythagoras. RT^2 = RS^2 + ST^2. So 5^2 = RS^2 + 2^2 => 25 = RS^2 + 4 => RS^2 = 21 => RS = sqrt(21). Which matches the coordinates given in the Asymptote code, since R is at (2, sqrt(21)), so RS is vertical from (2,0) to (2, sqrt(21)), so length sqrt(21). So that checks out.

Now, the question is asking for sin T. So angle T is at point T, which is the right angle? Wait no, angle T is at point T, which is connected to S and R. Since the right angle is at S, angle T is one of the acute angles. So in triangle RST, angle T is opposite to side RS. Wait, let me think again. In triangle RST, angle T is at point T. The sides opposite to angles are as follows: side opposite angle R is ST, side opposite angle S is RT, and side opposite angle T is RS. Therefore, sin T would be opposite over hypotenuse. The hypotenuse is RT, which is 5. The side opposite angle T is RS, which is sqrt(21). Therefore, sin T = RS / RT = sqrt(21)/5. But let me check if that's correct.

Alternatively, maybe I can use the coordinates to compute it. Since coordinates are given, maybe I can compute the lengths and then compute the sine. Let me confirm.

Coordinates:

T is at (0,0), S is at (2,0), R is at (2, sqrt(21)).

So, angle T is at (0,0). Let me find the sides adjacent and opposite to angle T. In triangle RST, angle T is between sides ST and RT. So, to find sin T, we can think of the opposite side over hypotenuse. Wait, but in a right-angled triangle, the hypotenuse is opposite the right angle. Since the right angle is at S, hypotenuse is RT. Therefore, for angle T, which is at (0,0), the sides are:

- Opposite side: RS, which is sqrt(21)

- Adjacent side: ST, which is 2

- Hypotenuse: RT, which is 5

Therefore, sin T = opposite / hypotenuse = RS / RT = sqrt(21)/5. Alternatively, if I think of angle T, then in terms of the triangle, sin T = opposite / hypotenuse. So that would be RS / RT. Which is sqrt(21)/5. So that would be the answer. But let me check if that's correct.

Alternatively, maybe I can compute it using coordinates. Let me compute the coordinates. The coordinates of T are (0,0), S is (2,0), R is (2, sqrt(21)). So vector from T to S is (2,0), and vector from T to R is (2, sqrt(21)). So angle at T is between vectors TS and TR. To find sin T, maybe I can compute the angle between those two vectors.

Alternatively, since we know the lengths of the sides, we can use trigonometry. Since we have a right-angled triangle, we can use the sides. Since angle T is adjacent to side ST (length 2) and opposite to side RS (length sqrt(21)), so sin T = opposite / hypotenuse = sqrt(21)/5. Therefore, the answer is sqrt(21)/5. But let me check if that's the case.

Alternatively, maybe I can use the fact that in a right-angled triangle, the two non-right angles are complementary. So angle R + angle T = 90 degrees. Therefore, sin T = sin(90 - R) = cos R. So if I can find cos R, then that would be sin T. Since sin R is 2/5, then cos R can be found using sin^2 R + cos^2 R = 1. Therefore, cos R = sqrt(1 - (2/5)^2) = sqrt(1 - 4/25) = sqrt(21/25) = sqrt(21)/5. Therefore, sin T = cos R = sqrt(21)/5. So that's another way to get the same answer. Therefore, sin T is sqrt(21)/5.

But let me check if that's consistent with the coordinates. Let me compute the coordinates. The coordinates of the triangle are T(0,0), S(2,0), R(2, sqrt(21)). So angle at T. Let me compute the lengths of the sides adjacent to angle T. The sides adjacent to angle T are ST and RT? Wait, no. In angle T, the sides are ST and RT. Wait, but in a triangle, the sides adjacent to angle T are the two sides that form the angle, which are ST and RT. However, in the right-angled triangle, the hypotenuse is RT, and the other sides are ST and RS. Wait, maybe I need to think again.

Wait, angle T is at point T. The sides forming angle T are ST and RT. The side opposite angle T is RS. So, in terms of trigonometry, sin T = opposite / hypotenuse = RS / RT = sqrt(21)/5. Which is the same as before. Alternatively, if I use coordinates, I can compute the angle. Let me compute the coordinates.

Vector TS is from T to S: (2,0) - (0,0) = (2,0). Vector TR is from T to R: (2, sqrt(21)) - (0,0) = (2, sqrt(21)). The angle between vectors TS and TR is angle T. To find sin T, we can use the cross product formula. The magnitude of the cross product of vectors TS and TR is |TS x TR| = |(2)(sqrt(21)) - (0)(2)| = 2*sqrt(21). The magnitude of TS is sqrt(2^2 + 0^2) = 2. The magnitude of TR is sqrt(2^2 + (sqrt(21))^2) = sqrt(4 + 21) = sqrt(25) = 5. Then, the sine of the angle between them is |TS x TR| / (|TS| |TR|) = 2*sqrt(21) / (2*5) = sqrt(21)/5. Which matches the previous results. Therefore, sin T is sqrt(21)/5. Therefore, the answer is sqrt(21)/5.

But let me check again if I didn't make any mistake. The problem says sin R is 2/5. Let me check that with coordinates. In triangle RST, angle R is at point R. The sides adjacent to angle R are RS and RT. Wait, no. In angle R, the sides are RS and RT. Wait, but in the triangle, angle R is between RS and RT. Wait, but since the triangle is right-angled at S, angle R is between RS and RT. Wait, but RS is vertical from S to R, and RT is the hypotenuse from R to T. So, to compute sin R, which is opposite over hypotenuse. The side opposite angle R is ST, which is 2, and hypotenuse is RT, which is 5. Therefore, sin R = 2/5, which matches the given information. Therefore, that's correct. Therefore, sin T is sqrt(21)/5. Therefore, the answer is sqrt(21)/5. So I think that's correct.

Alternatively, maybe I can use the area of the triangle. The area is (1/2)*ST*RS = (1/2)*2*sqrt(21) = sqrt(21). Also, area can be expressed as (1/2)*RT*height from S. Wait, but maybe not necessary. Alternatively, since we know all sides, maybe not needed.

Alternatively, since we have coordinates, we can compute the angle. For example, the coordinates of T(0,0), S(2,0), R(2, sqrt(21)). So, angle at T. The coordinates can be used to compute the slopes of the sides, but maybe that's more complicated. Alternatively, using trigonometry. Since we have the lengths, we can compute tan T. In triangle RST, tan T = opposite / adjacent = RS / ST = sqrt(21)/2. Therefore, sin T = tan T / sqrt(1 + tan^2 T) = [sqrt(21)/2] / sqrt(1 + (21/4)) = [sqrt(21)/2] / sqrt(25/4) = [sqrt(21)/2] / [5/2] = sqrt(21)/5. Which again gives the same result. Therefore, sin T is sqrt(21)/5.

Therefore, after multiple methods, I get the same answer. Therefore, I think that is correct. Therefore, the answer is sqrt(21)/5. So in boxed form, that would be \boxed{\dfrac{\sqrt{21}}{5}}.

**Final Answer**
\boxed{\dfrac{\sqrt{21}}{5}}

To determine $\sin T$ in triangle $RST$, we are given that $\sin R = \frac{2}{5}$ and that the triangle is right-angled at $S$. Let's proceed step by step using both geometric and trigonometric reasoning.

---

### Step 1: Understand the Triangle

From the Asymptote code and the description:

- $T = (0, 0)$
- $S = (2, 0)$
- $R = (2, \sqrt{21})$

This places the triangle in the coordinate plane with:

- $S$ at the origin,
- $T$ to the right of $S$,
- $R$ directly above $S$.

The right angle is at $S$, so triangle $RST$ is a **right triangle** with the right angle at $S$. Therefore:

- $RT$ is the **hypotenuse**,
- $RS$ and $ST$ are the **legs**.

---

### Step 2: Use Given Information

We are told that $\sin R = \frac{2}{5}$. In a right triangle, the sine of an angle is the ratio of the **opposite side** to the **hypotenuse**.

- Angle $R$ is at point $R$, so the side opposite to angle $R$ is $ST$.
- The hypotenuse is $RT$.

So:

$$
\sin R = \frac{ST}{RT} = \frac{2}{5}
$$

From this, we can deduce that:

$$
RT = 5 \quad \text{and} \quad ST = 2
$$

---

### Step 3: Use the Pythagorean Theorem to Find $RS$

Since triangle $RST$ is right-angled at $S$, we can apply the Pythagorean Theorem:

$$
RT^2 = RS^2 + ST^2
$$

$$
5^2 = RS^2 + 2^2 \Rightarrow 25 = RS^2 + 4 \Rightarrow RS^2 = 21 \Rightarrow RS = \sqrt{21}
$$

---

### Step 4: Find $\sin T$

Angle $T$ is at point $T$, and in triangle $RST$, the side opposite to angle $T$ is $RS$, and the hypotenuse is $RT$. So:

$$
\sin T = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{RS}{RT} = \frac{\sqrt{21}}{5}
$$

---

### Step 5: Confirm Using Complementary Angles

In a right triangle, the two non-right angles are **complementary**. That is:

$$
\angle R + \angle T = 90^\circ
$$

So:

$$
\sin T = \cos R
$$

We are given $\sin R = \frac{2}{5}$, so:

$$
\cos R = \sqrt{1 - \sin^2 R} = \sqrt{1 - \left(\frac{2}{5}\right)^2} = \sqrt{1 - \frac{4}{25}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}
$$

Thus, $\sin T = \cos R = \frac{\sqrt{21}}{5}$, confirming our earlier result.

---

### Final Answer

$$
\boxed{\dfrac{\sqrt{21}}{5}}
$$

Solution:
Because $\triangle RST$ is a right triangle, $\sin R = \frac{ST}{RT}$. So $\sin R = \frac{2}{5} = \frac{ST}{5}$. Then $ST=2$.

We know that $\sin T = \frac{RS}{RT}$. By the Pythagorean Theorem, $RS = \sqrt{RT^2 - ST^2} = \sqrt{25-4} = \sqrt{21}$. Then $\sin T = \boxed{\frac{\sqrt{21}}{5}}$.


Explanation:
Okay, so I need to figure out the ratio of the acceleration of Jen's softball to Jack's baseball, given that they both apply the same force. Let me start by recalling the formula for force. The problem mentions that force equals mass times acceleration, which is F = m * a. Right, so force is directly proportional to mass and acceleration.

Jen is throwing a softball, and Jack is throwing a baseball. They want to apply the same force. The question is asking for the ratio of their accelerations. So, I need to find a_jen / a_jack.

First, let me note down the given information. The mass of the softball is 200 grams, and the mass of the baseball is 150 grams. Wait, but in physics, we usually use kilograms for mass, right? But since both masses are given in grams, maybe I can just work with grams as long as I keep the units consistent. However, since the ratio will end up being unitless, maybe the actual units don't matter here. Let me think.

Since force is the same for both, F_jen = F_jack. Let me write that down.

F_jen = m_jen * a_jen
F_jack = m_jack * a_jack

Since F_jen = F_jack, then m_jen * a_jen = m_jack * a_jack.

So, if I want to find the ratio a_jen / a_jack, I can rearrange the equation. Let me do that.

Starting with m_jen * a_jen = m_jack * a_jack.

Divide both sides by m_jack * a_jen.

Wait, maybe it's better to solve for a_jen / a_jack. Let me rearrange the equation.

From m_jen * a_jen = m_jack * a_jack, divide both sides by m_jack * m_jen.

So, (a_jen / m_jack) = (a_jack / m_jen).

Wait, maybe another approach. Let me solve for a_jen and a_jack.

From F = m * a, we can write a = F / m.

Therefore, a_jen = F / m_jen
and a_jack = F / m_jack

Therefore, the ratio a_jen / a_jack is (F / m_jen) / (F / m_jack).

Simplify that. The F cancels out, so it becomes (1 / m_jen) * (m_jack / 1) = m_jack / m_jen.

Wait, so the ratio of accelerations is inverse of the ratio of masses? Because a_jen / a_jack = m_jack / m_jen.

So, if the masses are different, then the acceleration is inversely proportional to the mass when the force is the same.

So, let me check that again. If you have the same force, then a = F/m. So, if one mass is bigger, the acceleration is smaller. So, if Jen's softball is heavier than Jack's baseball, then her acceleration should be less than Jack's. Therefore, the ratio a_jen / a_jack should be less than 1.

Given that the mass of the softball is 200 grams and the baseball is 150 grams. So, m_jen = 200 g, m_jack = 150 g.

Therefore, the ratio a_jen / a_jack = m_jack / m_jen = 150 / 200.

Simplify that fraction. Both numerator and denominator can be divided by 50. 150 ÷ 50 = 3, 200 ÷ 50 = 4. So, 3/4.

Wait, so the ratio is 3/4? That would mean that Jen's acceleration is 3/4 of Jack's acceleration.

But let me make sure I didn't mix up the ratio. Let me go back.

a_jen = F / m_jen
a_jack = F / m_jack
Therefore, a_jen / a_jack = (F / m_jen) / (F / m_jack) = (F / m_jen) * (m_jack / F) = m_jack / m_jen.

Yes, that's correct. So, the ratio is m_jack / m_jen.

So, substituting the numbers, m_jack is 150 grams, m_jen is 200 grams.

So, 150 / 200 = 3/4.

Therefore, the ratio of acceleration of Jen's ball to Jack's is 3/4.

But let me think again. If the softball is heavier, then for the same force, it should have less acceleration. So, if Jen's acceleration is less than Jack's, then the ratio should be less than 1. Which 3/4 is. So that makes sense.

Alternatively, if I think of numbers. Suppose the force is, say, 1 N. Then, for the softball, acceleration is 1 N / 0.2 kg = 5 m/s². For the baseball, acceleration is 1 N / 0.15 kg ≈ 6.666... m/s². Then, the ratio of Jen's acceleration to Jack's is 5 / 6.666... = 0.75, which is 3/4.

Yes, that checks out.

So, converting grams to kilograms is important here. Wait, but in the calculation above, when I did 150 grams and 200 grams, I didn't convert to kg. But since both masses are in grams, they cancel out. Because when you take the ratio, the units are grams/grams, which is unitless. So, even if I convert them to kg, it would be 0.15 kg and 0.2 kg. Then, the ratio is 0.15 / 0.2 = 3/4. Same result.

So, regardless of the units, as long as they are consistent, the ratio is 3/4.

Therefore, the answer is 3/4.

But let me check once more.

Suppose force is same. So, F = m_jen * a_jen = m_jack * a_jack.

Therefore, a_jen / a_jack = m_jack / m_jen.

Yes. So, 150 / 200 = 3/4.

Yes.

Alternatively, if I think of it as, if you have the same force, then acceleration is inversely proportional to mass. So, the ratio of accelerations is inverse ratio of masses.

So, masses are 200 and 150. So, inverse ratio is 150:200, which simplifies to 3:4.

Therefore, the ratio of accelerations is 3:4.

So, the answer is 3/4.

I think that's solid.

**Final Answer**
The ratio of acceleration of Jen's ball to Jack's is \boxed{\dfrac{3}{4}}.

To determine the ratio of the acceleration of Jen's softball to Jack's baseball, we begin with the fundamental physics formula:

$$
F = m \cdot a
$$

Where:
- $ F $ is the force applied,
- $ m $ is the mass of the object,
- $ a $ is the resulting acceleration.

---

### Step 1: Express Acceleration in Terms of Force and Mass

From the formula, we can solve for acceleration:

$$
a = \frac{F}{m}
$$

Since both Jen and Jack apply the **same force**, we can write:

$$
a_{\text{Jen}} = \frac{F}{m_{\text{Jen}}}, \quad a_{\text{Jack}} = \frac{F}{m_{\text{Jack}}}
$$

---

### Step 2: Find the Ratio of Accelerations

We are asked to find the ratio of Jen's acceleration to Jack's:

$$
\frac{a_{\text{Jen}}}{a_{\text{Jack}}} = \frac{\frac{F}{m_{\text{Jen}}}}{\frac{F}{m_{\text{Jack}}}} = \frac{m_{\text{Jack}}}{m_{\text{Jen}}}
$$

This shows that the ratio of accelerations is **inversely proportional** to the ratio of the masses.

---

### Step 3: Plug in the Given Masses

We are given:
- $ m_{\text{Jen}} = 200 $ g (softball),
- $ m_{\text{Jack}} = 150 $ g (baseball).

So:

$$
\frac{a_{\text{Jen}}}{a_{\text{Jack}}} = \frac{150}{200}
$$

Simplify the fraction:

$$
\frac{150}{200} = \frac{3}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{3}{4}}
$$

This means that the acceleration of Jen's softball is **three-fourths** that of Jack's baseball, assuming they apply the same force.

[Assistant]
Explanation:
Okay, so I need to figure out the ratio of the acceleration of Jen's softball to Jack's baseball, given that they both apply the same force. Let me start by recalling the formula for force. The problem mentions that force equals mass times acceleration, which is F = m * a. Right, so force is directly proportional to mass and acceleration.

Jen is throwing a softball, and Jack is throwing a baseball. They want to apply the same force. The question is asking for the ratio of their accelerations. So, I need to find a_jen / a_jack.

First, let me note down the given information. The mass of the softball is 200 grams, and the mass of the baseball is 150 grams. Wait, but in physics, we usually use kilograms for mass, right? But since both masses are given in grams, maybe I can just work with grams as long as I keep the units consistent. However, since the ratio will end up being unitless, maybe the actual units don't matter here. Let me think.

Since force is the same for both, F_jen = F_jack. Let me write that down.

F_jen = m_jen * a_jen
F_jack = m_jack * a_jack

Since F_jen = F_jack, then m_jen * a_jen = m_jack * a_jack.

So, if I want to find the ratio a_jen / a_jack, I can rearrange the equation. Let me do that.

Starting with m_jen * a_jen = m_jack * a_jack.

Divide both sides by m_jack * a_jen.

Wait, maybe it's better to solve for a_jen / a_jack. Let me rearrange the equation.

From m_jen * a_jen = m_jack * a_jack, divide both sides by m_jack * m_jen.

So, (a_jen / m_jack) = (a_jack / m_jen).

Wait, maybe another approach. Let me solve for a_jen and a_jack.

From F = m * a, we can write a = F / m.

Therefore, a_jen = F / m_jen
and a_jack = F / m_jack

Therefore, the ratio a_jen / a_jack is (F / m_jen) / (F / m_jack).

Simplify that. The F cancels out, so it becomes (1 / m_jen) * (m_jack / 1) = m_jack / m_jen.

Wait, so the ratio of accelerations is inverse of the ratio of masses? Because a_jen / a_jack = m_jack / m_jen.

So, if the masses are different, then the acceleration is inversely proportional to the mass when the force is the same.

So, let me check that again. If you have the same force, then a = F/m. So, if one mass is bigger, the acceleration is smaller. So, if Jen's softball is heavier than Jack's baseball, then her acceleration should be less than Jack's. Therefore, the ratio a_jen / a_jack should be less than 1.

Given that the mass of the softball is 200 grams and the baseball is 150 grams. So, m_jen = 200 g, m_jack = 150 g.

Therefore, the ratio a_jen / a_jack = m_jack / m_jen = 150 / 200.

Simplify that fraction. Both numerator and denominator can be divided by 50. 150 ÷ 50 = 3, 200 ÷ 50 = 4. So, 3/4.

Wait, so the ratio is 3/4? That would mean that Jen's acceleration is 3/4 of Jack's acceleration.

But let me make sure I didn't mix up the ratio. Let me go back.

a_jen = F / m_jen
a_jack = F / m_jack
Therefore, a_jen / a_jack = (F / m_jen) / (F / m_jack) = (F / m_jen) * (m_jack / F) = m_jack / m_jen.

Yes, that's correct. So, the ratio is m_jack / m_jen.

So, substituting the numbers, m_jack is 150 grams, m_jen is 200 grams.

So, 150 / 200 = 3/4.

Therefore, the ratio of acceleration of Jen's ball to Jack's is 3/4.

But let me think again. If the softball is heavier, then for the same force, it should have less acceleration. So, if Jen's acceleration is less than Jack's, then the ratio should be less than 1. Which 3/4 is. So that makes sense.

Alternatively, if I think of numbers. Suppose the force is, say, 1 N. Then, for the softball, acceleration is 1 N / 0.2 kg = 5 m/s². For the baseball, acceleration is 1 N / 0.15 kg ≈ 6.666... m/s². Then, the ratio of Jen's acceleration to Jack's is 5 / 6.666... = 0.75, which is 3/4.

Yes, that checks out.

So, converting grams to kilograms is important here. Wait, but in the calculation above, when I did 150 grams and 200 grams, I didn't convert to kg. But since both masses are in grams, they cancel out. Because when you take the ratio, the units are grams/grams, which is unitless. So, even if I convert them to kg, it would be 0.15 kg and 0.2 kg. Then, the ratio is 0.15 / 0.2 = 3/4. Same result.

So, regardless of the units, as long as they are consistent, the ratio is 3/4.

Therefore, the answer is 3/4.

But let me check once more.

Suppose force is same. So, F = m_jen * a_jen = m_jack * a_jack.

Therefore, a_jen / a_jack = m_jack / m_jen.

Yes. So, 150 / 200 = 3/4.

Yes.

Alternatively, if I think of it as, if you have the same force, then acceleration is inversely proportional to mass. So, the ratio of accelerations is inverse ratio of masses.

So, masses are 200 and 150. So, inverse ratio is 150:200, which simplifies to 3:4.

Therefore, the ratio of accelerations is 3:4.

So, the answer is 3/4.

I think that's solid.

**Final Answer**
The ratio of acceleration of Jen's ball to Jack's is \boxed{\dfrac{3}{4}}.

To determine the ratio of the acceleration of Jen's softball to Jack's baseball, we begin with the fundamental physics formula:

$$
F = m \cdot a
$$

Where:
- $ F $ is the force applied,
- $ m $ is the mass of the object,
- $ a $ is the resulting acceleration.

---

### Step 1: Express Acceleration in Terms of Force and Mass

From the formula, we can solve for acceleration:

$$
a = \frac{F}{m}
$$

Since both Jen and Jack apply the **same force**, we can write:

$$
a_{\text{Jen}} = \frac{F}{m_{\text{Jen}}}, \quad a_{\text{Jack}} = \frac{F}{m_{\text{Jack}}}
$$

---

### Step 2: Find the Ratio of Accelerations

We are asked to find the ratio of Jen's acceleration to Jack's:

$$
\frac{a_{\text{Jen}}}{a_{\text{Jack}}} = \frac{\frac{F}{m_{\text{Jen}}}}{\frac{F}{m_{\text{Jack}}}} = \frac{m_{\text{Jack}}}{m_{\text{Jen}}}
$$

This shows that the ratio of accelerations is **inversely proportional** to the ratio of the masses.

---

### Step 3: Plug in the Given Masses

We are given:
- $ m_{\text{Jen}} = 200 $ g (softball),
- $ m_{\text{Jack}} = 150 $ g (baseball).

So:

$$
\frac{a_{\text{Jen}}}{a_{\text{Jack}}} = \frac{150}{200}
$$

Simplify the fraction:

$$
\frac{150}{200} = \frac{3}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{3}{4}}
$$

This means that the acceleration of Jen's softball is **three-fourths** that of Jack's baseball, assuming they apply the same force.

Solution: If $j_1$ is the acceleration of Jen's ball and $j_2$ is the acceleration of Jack's ball, then we have $$j_1 \cdot 200 = j_2 \cdot 150\qquad \Rightarrow\qquad \frac{j_1}{j_2} = \boxed{\frac 34}.$$