Explanation:
Okay, so I need to find the smallest distance between the origin (0,0) and a point on the graph of the function y = (1/2)x² - 9. Then, once I find that smallest distance, which they call 'a', I have to find a². Alright, let me think about how to approach this.

First, I remember that the distance between two points (x1, y1) and (x2, y2) is given by the distance formula: √[(x2 - x1)² + (y2 - y1)²]. In this case, one of the points is the origin, so (x1, y1) is (0, 0), and the other point is (x, y) which lies on the graph of the given function. Therefore, the distance from the origin to a point (x, y) on the graph is √[(x - 0)² + (y - 0)²] = √(x² + y²).

But since the point (x, y) is on the graph of y = (1/2)x² - 9, I can substitute y in the distance formula with that expression. So, substituting, the distance becomes √[x² + ((1/2)x² - 9)²]. Therefore, the distance squared would be x² + ((1/2)x² - 9)². Since the square root function is increasing, the minimum distance occurs at the same x-value as the minimum of the distance squared. So, maybe it's easier to minimize the distance squared instead of the distance itself to avoid dealing with the square root. That should simplify calculations.

So, let me define D(x) = x² + [(1/2)x² - 9]^2. I need to find the minimum value of D(x). To find the minimum, I can take the derivative of D(x) with respect to x, set it equal to zero, and solve for x. Then, check if it's a minimum using the second derivative or some other method.

Let me compute D(x) first. Let me expand [(1/2)x² - 9]^2. Let me do that step by step. Let me denote u = (1/2)x² - 9. Then, u² = [(1/2)x² - 9]^2. Expanding that, it's ( (1/2)x² )² - 2*(1/2)x²*9 + 9². Let me compute each term:

First term: ( (1/2)x² )² = (1/4)x⁴.

Second term: -2*(1/2)x²*9 = - (2*(1/2)*9)x² = -9x².

Third term: 9² = 81.

Therefore, [(1/2)x² - 9]^2 = (1/4)x⁴ - 9x² + 81.

Therefore, D(x) = x² + (1/4)x⁴ - 9x² + 81. Let me combine like terms:

x² - 9x² is -8x². So, D(x) = (1/4)x⁴ - 8x² + 81.

So, D(x) = (1/4)x⁴ - 8x² + 81. Now, to find the minimum, take derivative D’(x):

D’(x) = d/dx [ (1/4)x⁴ - 8x² + 81 ] = (1/4)*4x³ - 8*2x + 0 = x³ - 16x.

Set derivative equal to zero:

x³ - 16x = 0.

Factor:

x(x² - 16) = 0.

Which factors further as x(x - 4)(x + 4) = 0. Therefore, critical points at x = 0, x = 4, x = -4.

Now, I need to check which of these critical points gives the minimum distance. Since the problem is about the smallest distance, and the graph is a parabola opening upwards, I need to check these points.

First, let me compute D(x) at each critical point.

Starting with x = 0:

D(0) = (1/4)(0)^4 - 8*(0)^2 + 81 = 0 - 0 + 81 = 81. Therefore, distance squared is 81, so distance is 9. But let me check if that's the minimum.

Next, x = 4:

D(4) = (1/4)*(4)^4 - 8*(4)^2 + 81.

Compute each term:

(1/4)*(256) = 64.

-8*(16) = -128.

So, 64 - 128 + 81 = (64 + 81) - 128 = 145 - 128 = 17. So D(4) = 17. That's much smaller than 81. Similarly, for x = -4, since the function D(x) is even? Let me check.

Wait, D(x) is (1/4)x⁴ - 8x² + 81. Since x⁴ and x² are even functions, D(-x) = D(x). Therefore, D(-4) = D(4) = 17. So both x = 4 and x = -4 give the same distance squared of 17. So, that seems like a candidate for minimum.

But let me check if there are any other critical points? Well, we found all critical points by solving x³ - 16x = 0, which gave x = 0, 4, -4. So, those are all. So, the minimum occurs at x = ±4, with D(x) = 17. Therefore, the minimal distance squared is 17, so a² is 17? Wait, but hold on. Wait, the problem says "the smallest distance between the origin and a point on the graph... can be expressed as a. Find a²." So, if the minimal distance squared is 17, then a is √17, so a² is 17. Therefore, the answer is 17? Wait, but let me verify this because sometimes when you minimize distance squared, you have to make sure that it's indeed the minimum.

Alternatively, maybe I made a mistake in calculation? Let me check again.

First, D(x) = x² + [(1/2)x² - 9]^2. Let me compute D(4):

First, compute y when x = 4: y = (1/2)(16) - 9 = 8 - 9 = -1. Therefore, the point is (4, -1). Then, distance squared is 4² + (-1)^2 = 16 + 1 = 17. Correct. Similarly, for x = -4, same thing: (-4)^2 + (-1)^2 = 16 + 1 = 17. So that's correct.

For x = 0, the point is (0, -9). Distance squared is 0² + (-9)^2 = 81. Which is larger. So, yes, 17 is smaller. So, seems like 17 is the minimal distance squared. Therefore, a² is 17. Therefore, the answer is 17. Wait, but let me check if there are any other points where the distance could be smaller?

Alternatively, maybe I can approach this problem using calculus, but also maybe using geometry? For example, the minimal distance from a point to a curve is along the line perpendicular to the tangent of the curve at that point. So, maybe I can use that method as a check.

Let me try that approach. Let me recall that the shortest distance from a point to a curve occurs where the line connecting the point to the curve is perpendicular to the tangent of the curve at that point.

So, let me consider a general point (x, y) on the curve y = (1/2)x² - 9. The slope of the tangent line to the curve at that point is dy/dx = derivative of (1/2)x² - 9, which is x. Therefore, the slope of the tangent line is m_tangent = x. Therefore, the slope of the line perpendicular to the tangent (i.e., the line connecting the origin to the point (x, y)) should be -1/m_tangent, provided that m_tangent ≠ 0. So, slope of the line connecting (0, 0) to (x, y) is (y - 0)/(x - 0) = y/x. Therefore, according to the condition, this slope should be equal to -1/m_tangent = -1/x. Therefore, we have y/x = -1/x. Wait, but that would imply y = -1. Wait, but y is (1/2)x² - 9. Therefore, setting y = -1:

(1/2)x² - 9 = -1 => (1/2)x² = 8 => x² = 16 => x = ±4. Which is exactly the same result as before! Therefore, the points are (4, -1) and (-4, -1), which gives the minimal distance squared of 17. Therefore, this confirms the previous result. Therefore, a² is 17. Therefore, the answer is 17. So, I think that's correct.

But just to be thorough, let me check if there are any other critical points or if there's a possibility that the minimal distance occurs at some other point. For example, maybe when the derivative is undefined? But the function D(x) is a polynomial, so it's differentiable everywhere. Therefore, the only critical points are at x = 0, 4, -4. So, we have checked all of them. Therefore, 17 is indeed the minimal distance squared.

Alternatively, maybe I can use another method, like parametric equations or Lagrange multipliers? Let me try Lagrange multipliers for practice.

We need to minimize the distance squared function f(x, y) = x² + y², subject to the constraint g(x, y) = y - (1/2)x² + 9 = 0.

The Lagrangian is L(x, y, λ) = x² + y² - λ(y - (1/2)x² + 9).

Taking partial derivatives:

∂L/∂x = 2x - λ*(-x) = 2x + λx = 0.

Wait, let me compute that again. Wait, the constraint is g(x, y) = y - (1/2)x² + 9 = 0. Therefore, the gradient of g is (∂g/∂x, ∂g/∂y) = (-x, 1). Therefore, the Lagrangian equations are:

∇f = λ∇g.

So, ∇f = (2x, 2y).

Therefore, equations:

2x = λ*(-x)  --> 2x = -λx

2y = λ*(1) --> 2y = λ

And the constraint y - (1/2)x² + 9 = 0.

So, from the first equation: 2x = -λx. Let's rearrange:

2x + λx = 0 => x(2 + λ) = 0. Therefore, either x = 0 or λ = -2.

Case 1: x = 0.

Then, from the constraint equation: y - (1/2)(0)^2 + 9 = y + 9 = 0 => y = -9. Therefore, the point is (0, -9). Then, distance squared is 0² + (-9)^2 = 81, which we already saw.

Case 2: λ = -2.

From the second equation: 2y = λ = -2 => y = -1.

Then, from the constraint equation: y - (1/2)x² + 9 = -1 - (1/2)x² + 9 = 8 - (1/2)x² = 0 => (1/2)x² = 8 => x² = 16 => x = ±4. Therefore, points (4, -1) and (-4, -1). Then, distance squared is 16 + 1 = 17. So, same result as before. Therefore, this method also gives the same answer. Therefore, this confirms that the minimal distance squared is indeed 17.

Therefore, I think that's solid. Therefore, the answer is 17. Therefore, a² is 17. Therefore, the answer is \boxed{17}.

But just to make sure, let me think if there's another way someone might approach this problem and maybe get confused. For example, someone might think to parametrize the distance function and try to minimize it, but that's essentially what I did. Alternatively, someone might try to use geometry, like finding the closest point on the parabola to the origin. But since the parabola opens upwards, the closest point is likely to be somewhere below the vertex? Wait, the vertex of the parabola y = (1/2)x² - 9 is at (0, -9). So, the vertex is the lowest point on the parabola. The origin is at (0, 0). So, the distance from the origin to the vertex is sqrt(0² + (-9 - 0)^2) = 9, which is the distance we saw for x=0. But since the parabola extends upwards, maybe there's a point on the parabola that is closer to the origin? Which is exactly what we found: points (4, -1) and (-4, -1) are closer. So, the distance from the origin to those points is sqrt(17) ≈ 4.123, which is less than 9. So, that makes sense.

Alternatively, someone might think that the closest point is the vertex, but that's not the case here. So, that's a common mistake. Therefore, it's important to check.

Alternatively, maybe someone could graph the function and see? But since I can't graph here, but I can reason. The parabola is opening upwards, vertex at (0, -9). The origin is above the vertex. So, the closest point might be somewhere on the parabola where the line from the origin to the point is perpendicular to the tangent, which we found at x=±4. So, that seems correct.

Alternatively, maybe using calculus, but I think I did that already. So, all methods lead to the same answer. Therefore, I think 17 is correct.

Therefore, the answer is 17. So, a² is 17. Therefore, boxed answer is \boxed{17}.

**Final Answer**
\boxed{17}

To find the smallest distance between the origin $(0, 0)$ and a point on the graph of the function
$$
y = \frac{1}{2}x^2 - 9,
$$
we begin by using the **distance formula** between two points $(x, y)$ and $(0, 0)$:

$$
\text{Distance} = \sqrt{x^2 + y^2}.
$$

Since the point $(x, y)$ lies on the graph of the function, we substitute $y = \frac{1}{2}x^2 - 9$ into the distance formula:

$$
\text{Distance} = \sqrt{x^2 + \left(\frac{1}{2}x^2 - 9\right)^2}.
$$

To simplify the minimization process, we consider the **square of the distance**, which is easier to work with:

$$
D(x) = x^2 + \left(\frac{1}{2}x^2 - 9\right)^2.
$$

---

### Step 1: Expand the expression

We expand the square term:

$$
\left(\frac{1}{2}x^2 - 9\right)^2 = \left(\frac{1}{2}x^2\right)^2 - 2 \cdot \frac{1}{2}x^2 \cdot 9 + 9^2 = \frac{1}{4}x^4 - 9x^2 + 81.
$$

So the distance squared becomes:

$$
D(x) = x^2 + \left(\frac{1}{4}x^4 - 9x^2 + 81\right) = \frac{1}{4}x^4 - 8x^2 + 81.
$$

---

### Step 2: Minimize the function

To find the minimum, we take the derivative of $D(x)$:

$$
D'(x) = \frac{d}{dx} \left( \frac{1}{4}x^4 - 8x^2 + 81 \right) = x^3 - 16x.
$$

Set the derivative equal to zero to find critical points:

$$
x^3 - 16x = 0 \Rightarrow x(x^2 - 16) = 0 \Rightarrow x = 0, \pm 4.
$$

---

### Step 3: Evaluate the distance squared at critical points

We now evaluate $D(x)$ at these critical points:

- At $x = 0$:
  $$
  D(0) = \frac{1}{4}(0)^4 - 8(0)^2 + 81 = 81.
  $$

- At $x = 4$:
  $$
  D(4) = \frac{1}{4}(256) - 8(16) + 81 = 64 - 128 + 81 = 17.
  $$

- At $x = -4$:
  $$
  D(-4) = \frac{1}{4}(256) - 8(16) + 81 = 64 - 128 + 81 = 17.
  $$

So the **minimum distance squared** is $17$, which occurs at $x = \pm 4$.

---

### Step 4: Interpret the result

Let $a$ be the smallest distance from the origin to the graph. Then:

$$
a^2 = \text{minimum of } D(x) = 17.
$$

---

### Final Answer

$$
\boxed{17}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest distance between the origin (0,0) and a point on the graph of the function y = (1/2)x² - 9. Then, once I find that smallest distance, which they call 'a', I have to find a². Alright, let me think about how to approach this.

First, I remember that the distance between two points (x1, y1) and (x2, y2) is given by the distance formula: √[(x2 - x1)² + (y2 - y1)²]. In this case, one of the points is the origin, so (x1, y1) is (0, 0), and the other point is (x, y) which lies on the graph of the given function. Therefore, the distance from the origin to a point (x, y) on the graph is √[(x - 0)² + (y - 0)²] = √(x² + y²).

But since the point (x, y) is on the graph of y = (1/2)x² - 9, I can substitute y in the distance formula with that expression. So, substituting, the distance becomes √[x² + ((1/2)x² - 9)²]. Therefore, the distance squared would be x² + ((1/2)x² - 9)². Since the square root function is increasing, the minimum distance occurs at the same x-value as the minimum of the distance squared. So, maybe it's easier to minimize the distance squared instead of the distance itself to avoid dealing with the square root. That should simplify calculations.

So, let me define D(x) = x² + [(1/2)x² - 9]^2. I need to find the minimum value of D(x). To find the minimum, I can take the derivative of D(x) with respect to x, set it equal to zero, and solve for x. Then, check if it's a minimum using the second derivative or some other method.

Let me compute D(x) first. Let me expand [(1/2)x² - 9]^2. Let me do that step by step. Let me denote u = (1/2)x² - 9. Then, u² = [(1/2)x² - 9]^2. Expanding that, it's ( (1/2)x² )² - 2*(1/2)x²*9 + 9². Let me compute each term:

First term: ( (1/2)x² )² = (1/4)x⁴.

Second term: -2*(1/2)x²*9 = - (2*(1/2)*9)x² = -9x².

Third term: 9² = 81.

Therefore, [(1/2)x² - 9]^2 = (1/4)x⁴ - 9x² + 81.

Therefore, D(x) = x² + (1/4)x⁴ - 9x² + 81. Let me combine like terms:

x² - 9x² is -8x². So, D(x) = (1/4)x⁴ - 8x² + 81.

So, D(x) = (1/4)x⁴ - 8x² + 81. Now, to find the minimum, take derivative D’(x):

D’(x) = d/dx [ (1/4)x⁴ - 8x² + 81 ] = (1/4)*4x³ - 8*2x + 0 = x³ - 16x.

Set derivative equal to zero:

x³ - 16x = 0.

Factor:

x(x² - 16) = 0.

Which factors further as x(x - 4)(x + 4) = 0. Therefore, critical points at x = 0, x = 4, x = -4.

Now, I need to check which of these critical points gives the minimum distance. Since the problem is about the smallest distance, and the graph is a parabola opening upwards, I need to check these points.

First, let me compute D(x) at each critical point.

Starting with x = 0:

D(0) = (1/4)(0)^4 - 8*(0)^2 + 81 = 0 - 0 + 81 = 81. Therefore, distance squared is 81, so distance is 9. But let me check if that's the minimum.

Next, x = 4:

D(4) = (1/4)*(4)^4 - 8*(4)^2 + 81.

Compute each term:

(1/4)*(256) = 64.

-8*(16) = -128.

So, 64 - 128 + 81 = (64 + 81) - 128 = 145 - 128 = 17. So D(4) = 17. That's much smaller than 81. Similarly, for x = -4, since the function D(x) is even? Let me check.

Wait, D(x) is (1/4)x⁴ - 8x² + 81. Since x⁴ and x² are even functions, D(-x) = D(x). Therefore, D(-4) = D(4) = 17. So both x = 4 and x = -4 give the same distance squared of 17. So, that seems like a candidate for minimum.

But let me check if there are any other critical points? Well, we found all critical points by solving x³ - 16x = 0, which gave x = 0, 4, -4. So, those are all. So, the minimum occurs at x = ±4, with D(x) = 17. Therefore, the minimal distance squared is 17, so a² is 17? Wait, but hold on. Wait, the problem says "the smallest distance between the origin and a point on the graph... can be expressed as a. Find a²." So, if the minimal distance squared is 17, then a is √17, so a² is 17. Therefore, the answer is 17? Wait, but let me verify this because sometimes when you minimize distance squared, you have to make sure that it's indeed the minimum.

Alternatively, maybe I made a mistake in calculation? Let me check again.

First, D(x) = x² + [(1/2)x² - 9]^2. Let me compute D(4):

First, compute y when x = 4: y = (1/2)(16) - 9 = 8 - 9 = -1. Therefore, the point is (4, -1). Then, distance squared is 4² + (-1)^2 = 16 + 1 = 17. Correct. Similarly, for x = -4, same thing: (-4)^2 + (-1)^2 = 16 + 1 = 17. So that's correct.

For x = 0, the point is (0, -9). Distance squared is 0² + (-9)^2 = 81. Which is larger. So, yes, 17 is smaller. So, seems like 17 is the minimal distance squared. Therefore, a² is 17. Therefore, the answer is 17. Wait, but let me check if there are any other points where the distance could be smaller?

Alternatively, maybe I can approach this problem using calculus, but also maybe using geometry? For example, the minimal distance from a point to a curve is along the line perpendicular to the tangent of the curve at that point. So, maybe I can use that method as a check.

Let me try that approach. Let me recall that the shortest distance from a point to a curve occurs where the line connecting the point to the curve is perpendicular to the tangent of the curve at that point.

So, let me consider a general point (x, y) on the curve y = (1/2)x² - 9. The slope of the tangent line to the curve at that point is dy/dx = derivative of (1/2)x² - 9, which is x. Therefore, the slope of the tangent line is m_tangent = x. Therefore, the slope of the line perpendicular to the tangent (i.e., the line connecting the origin to the point (x, y)) should be -1/m_tangent, provided that m_tangent ≠ 0. So, slope of the line connecting (0, 0) to (x, y) is (y - 0)/(x - 0) = y/x. Therefore, according to the condition, this slope should be equal to -1/m_tangent = -1/x. Therefore, we have y/x = -1/x. Wait, but that would imply y = -1. Wait, but y is (1/2)x² - 9. Therefore, setting y = -1:

(1/2)x² - 9 = -1 => (1/2)x² = 8 => x² = 16 => x = ±4. Which is exactly the same result as before! Therefore, the points are (4, -1) and (-4, -1), which gives the minimal distance squared of 17. Therefore, this confirms the previous result. Therefore, a² is 17. Therefore, the answer is 17. So, I think that's correct.

But just to be thorough, let me check if there are any other critical points or if there's a possibility that the minimal distance occurs at some other point. For example, maybe when the derivative is undefined? But the function D(x) is a polynomial, so it's differentiable everywhere. Therefore, the only critical points are at x = 0, 4, -4. So, we have checked all of them. Therefore, 17 is indeed the minimal distance squared.

Alternatively, maybe I can use another method, like parametric equations or Lagrange multipliers? Let me try Lagrange multipliers for practice.

We need to minimize the distance squared function f(x, y) = x² + y², subject to the constraint g(x, y) = y - (1/2)x² + 9 = 0.

The Lagrangian is L(x, y, λ) = x² + y² - λ(y - (1/2)x² + 9).

Taking partial derivatives:

∂L/∂x = 2x - λ*(-x) = 2x + λx = 0.

Wait, let me compute that again. Wait, the constraint is g(x, y) = y - (1/2)x² + 9 = 0. Therefore, the gradient of g is (∂g/∂x, ∂g/∂y) = (-x, 1). Therefore, the Lagrangian equations are:

∇f = λ∇g.

So, ∇f = (2x, 2y).

Therefore, equations:

2x = λ*(-x)  --> 2x = -λx

2y = λ*(1) --> 2y = λ

And the constraint y - (1/2)x² + 9 = 0.

So, from the first equation: 2x = -λx. Let's rearrange:

2x + λx = 0 => x(2 + λ) = 0. Therefore, either x = 0 or λ = -2.

Case 1: x = 0.

Then, from the constraint equation: y - (1/2)(0)^2 + 9 = y + 9 = 0 => y = -9. Therefore, the point is (0, -9). Then, distance squared is 0² + (-9)^2 = 81, which we already saw.

Case 2: λ = -2.

From the second equation: 2y = λ = -2 => y = -1.

Then, from the constraint equation: y - (1/2)x² + 9 = -1 - (1/2)x² + 9 = 8 - (1/2)x² = 0 => (1/2)x² = 8 => x² = 16 => x = ±4. Therefore, points (4, -1) and (-4, -1). Then, distance squared is 16 + 1 = 17. So, same result as before. Therefore, this method also gives the same answer. Therefore, this confirms that the minimal distance squared is indeed 17.

Therefore, I think that's solid. Therefore, the answer is 17. Therefore, a² is 17. Therefore, the answer is \boxed{17}.

But just to make sure, let me think if there's another way someone might approach this problem and maybe get confused. For example, someone might think to parametrize the distance function and try to minimize it, but that's essentially what I did. Alternatively, someone might try to use geometry, like finding the closest point on the parabola to the origin. But since the parabola opens upwards, the closest point is likely to be somewhere below the vertex? Wait, the vertex of the parabola y = (1/2)x² - 9 is at (0, -9). So, the vertex is the lowest point on the parabola. The origin is at (0, 0). So, the distance from the origin to the vertex is sqrt(0² + (-9 - 0)^2) = 9, which is the distance we saw for x=0. But since the parabola extends upwards, maybe there's a point on the parabola that is closer to the origin? Which is exactly what we found: points (4, -1) and (-4, -1) are closer. So, the distance from the origin to those points is sqrt(17) ≈ 4.123, which is less than 9. So, that makes sense.

Alternatively, someone might think that the closest point is the vertex, but that's not the case here. So, that's a common mistake. Therefore, it's important to check.

Alternatively, maybe someone could graph the function and see? But since I can't graph here, but I can reason. The parabola is opening upwards, vertex at (0, -9). The origin is above the vertex. So, the closest point might be somewhere on the parabola where the line from the origin to the point is perpendicular to the tangent, which we found at x=±4. So, that seems correct.

Alternatively, maybe using calculus, but I think I did that already. So, all methods lead to the same answer. Therefore, I think 17 is correct.

Therefore, the answer is 17. So, a² is 17. Therefore, boxed answer is \boxed{17}.

**Final Answer**
\boxed{17}

To find the smallest distance between the origin $(0, 0)$ and a point on the graph of the function
$$
y = \frac{1}{2}x^2 - 9,
$$
we begin by using the **distance formula** between two points $(x, y)$ and $(0, 0)$:

$$
\text{Distance} = \sqrt{x^2 + y^2}.
$$

Since the point $(x, y)$ lies on the graph of the function, we substitute $y = \frac{1}{2}x^2 - 9$ into the distance formula:

$$
\text{Distance} = \sqrt{x^2 + \left(\frac{1}{2}x^2 - 9\right)^2}.
$$

To simplify the minimization process, we consider the **square of the distance**, which is easier to work with:

$$
D(x) = x^2 + \left(\frac{1}{2}x^2 - 9\right)^2.
$$

---

### Step 1: Expand the expression

We expand the square term:

$$
\left(\frac{1}{2}x^2 - 9\right)^2 = \left(\frac{1}{2}x^2\right)^2 - 2 \cdot \frac{1}{2}x^2 \cdot 9 + 9^2 = \frac{1}{4}x^4 - 9x^2 + 81.
$$

So the distance squared becomes:

$$
D(x) = x^2 + \left(\frac{1}{4}x^4 - 9x^2 + 81\right) = \frac{1}{4}x^4 - 8x^2 + 81.
$$

---

### Step 2: Minimize the function

To find the minimum, we take the derivative of $D(x)$:

$$
D'(x) = \frac{d}{dx} \left( \frac{1}{4}x^4 - 8x^2 + 81 \right) = x^3 - 16x.
$$

Set the derivative equal to zero to find critical points:

$$
x^3 - 16x = 0 \Rightarrow x(x^2 - 16) = 0 \Rightarrow x = 0, \pm 4.
$$

---

### Step 3: Evaluate the distance squared at critical points

We now evaluate $D(x)$ at these critical points:

- At $x = 0$:
  $$
  D(0) = \frac{1}{4}(0)^4 - 8(0)^2 + 81 = 81.
  $$

- At $x = 4$:
  $$
  D(4) = \frac{1}{4}(256) - 8(16) + 81 = 64 - 128 + 81 = 17.
  $$

- At $x = -4$:
  $$
  D(-4) = \frac{1}{4}(256) - 8(16) + 81 = 64 - 128 + 81 = 17.
  $$

So the **minimum distance squared** is $17$, which occurs at $x = \pm 4$.

---

### Step 4: Interpret the result

Let $a$ be the smallest distance from the origin to the graph. Then:

$$
a^2 = \text{minimum of } D(x) = 17.
$$

---

### Final Answer

$$
\boxed{17}
$$

Solution:
By the distance formula, we are trying to minimize $\sqrt{x^2+y^2}=\sqrt{x^2+\frac{1}{4}x^4-9x^2+81}$. In general, minimization problems like this require calculus, but one optimization method that sometimes works is to try to complete the square.  Pulling out a factor of $\frac{1}{4}$ from under the radical, we have \begin{align*}
\frac{1}{2}\sqrt{4x^2+x^4-36x^2+324}&=\frac{1}{2}\sqrt{(x^4-32x^2+256)+68} \\
&= \frac{1}{2}\sqrt{(x^2-16)^2+68}
\end{align*}This last expression is minimized when the square equals $0$, that is, when $x^2=16$. Then the distance is $\frac{\sqrt{68}}{2}=\sqrt{17}$.  Hence the desired answer is $\sqrt{17}^2 = \boxed{17}$.


Explanation:
Okay, so I need to figure out how much money will be in the investment after five years if it's compounded bi-monthly at 1% interest. Let me start by understanding the problem step by step.

First, the initial investment is $24,000. The interest is 1% every two months. That means every two months, the investment increases by 1% of its current value. So, this is compound interest, right? Because each time, the interest is calculated on the current amount, not just the original principal.

Now, the question is asking for the total amount after five years. So, I need to figure out how many bi-monthly periods there are in five years. Let me recall that "bi-monthly" means every two months. So, how many two-month periods are in a year? Well, a year has 12 months, so 12 divided by 2 is 6. Therefore, there are 6 bi-monthly periods in a year. Therefore, over five years, that would be 5 times 6, which is 30 periods. Let me check that again. If each period is two months, then 5 years is 60 months. Divided by 2, that's 30 periods. Yep, that seems right.

So, the formula for compound interest is A = P(1 + r/n)^(nt), but wait, in this case, the compounding is happening every two months, so maybe I need to adjust the formula accordingly. Wait, actually, since the interest is compounded every two months, the formula would be A = P(1 + r)^t, where r is the interest rate per period, and t is the number of periods. Let me confirm.

In standard compound interest, A = P(1 + r/n)^(nt), where n is the number of times compounded per year. But here, the compounding is every two months, so if we consider the rate per period, it's 1% per period, and the number of periods is 30. So, maybe it's simpler. Let me think.

Alternatively, since the interest is 1% every two months, then each period, the amount is multiplied by 1.01. Therefore, after t periods, the amount is P*(1.01)^t. So, yes, that seems straightforward. So, in this case, P is 24,000, t is 30, and the rate per period is 1%, so 0.01. Therefore, A = 24000*(1.01)^30. Then, compute that value and round to the nearest whole number.

But let me make sure I didn't make a mistake here. Let me check the formula again. If it's compounded every two months, then the formula is indeed A = P*(1 + r)^t, where r is the rate per period, and t is the number of periods. So, since each period is two months, and the rate is 1% per period, then yes, that formula applies. So, that seems correct.

Alternatively, if I wanted to use the standard formula, I could think of it as annual compounding, but since it's bi-monthly, maybe not. Let me think. If I wanted to use the standard formula, I would need to know the annual interest rate. But here, the problem states that the interest is 1% bi-monthly. So, that is, every two months, the interest is 1%. Therefore, the rate per period is 1%, and the number of periods is 30. Therefore, the formula is correct.

So, moving forward with that. Let me compute 24000*(1.01)^30. Now, I need to calculate (1.01)^30. Let me recall that (1 + r)^t can be calculated using logarithms or using the formula for compound interest. Alternatively, I can use the rule of 72 to estimate, but since I need an exact value, I need to calculate it more precisely.

Alternatively, I can use the formula for compound interest. Let me try to compute (1.01)^30. Let me remember that ln(1.01) is approximately 0.00995, so multiplying by 30 gives approximately 0.2985. Then, exponentiating gives e^0.2985 ≈ 1.347. But this is an approximation. However, since I need an exact value, maybe I should use a calculator? Wait, but since this is a problem-solving scenario, perhaps I need to compute it step by step or use a better approximation.

Alternatively, maybe I can use the binomial theorem? But that might be too tedious. Alternatively, I can use the fact that (1.01)^30 is equal to e^(30*ln(1.01)). Let me compute ln(1.01). Let me recall that ln(1.01) is approximately 0.00995033. Let me check with more precision. Let me remember that ln(1+x) ≈ x - x^2/2 + x^3/3 - x^4/4 + ... for small x. Here, x is 0.01, so ln(1.01) ≈ 0.01 - 0.00005 + 0.00000333 - ... which is approximately 0.00995033. So, multiplying by 30 gives 0.2985099. Then, e^0.2985099. Let me compute e^0.2985.

We know that e^0.3 is approximately 1.349858. Since 0.2985 is slightly less than 0.3, so e^0.2985 ≈ e^(0.3 - 0.0015) = e^0.3 * e^(-0.0015). We know e^(-0.0015) ≈ 1 - 0.0015 + ... ≈ 0.9985. Therefore, e^0.2985 ≈ 1.349858 * 0.9985 ≈ 1.349858 - 1.349858*0.0015 ≈ 1.349858 - 0.002024787 ≈ 1.34783. Therefore, approximately 1.3478. Therefore, (1.01)^30 ≈ 1.3478. Therefore, multiplying by 24,000 gives 24,000 * 1.3478 ≈ ?

Let me compute 24,000 * 1.3478. Let me break it down:

First, 24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.0078 = ?

Wait, 1.3478 is 1 + 0.3 + 0.04 + 0.0078. Let me check:

1 + 0.3 = 1.3

1.3 + 0.04 = 1.34

1.34 + 0.0078 = 1.3478. Yes.

So, 24,000 * 1.3478 = 24,000 * 1 + 24,000 * 0.3 + 24,000 * 0.04 + 24,000 * 0.0078.

So, 24,000 + 7,200 + 960 + ?

Compute 24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

Now, 24,000 * 0.0078. Let me compute 24,000 * 0.0078.

First, 24,000 * 0.007 = 168

24,000 * 0.0008 = 19.2

So, total is 168 + 19.2 = 187.2

Therefore, total amount is 32,160 + 187.2 = 32,347.2

So, approximately $32,347.20. But this is based on my approximation of (1.01)^30 ≈ 1.3478. However, I need to check if this approximation is accurate enough. Because if my estimation of (1.01)^30 is off, then my final answer will be off.

Alternatively, maybe I can calculate (1.01)^30 more accurately. Let me try to compute it step by step. Let me recall that (1.01)^10 is approximately 1.104622, and then (1.104622)^3 would be (1.01)^30. Let me check that.

First, compute (1.01)^10. Let me compute that:

(1.01)^1 = 1.01

(1.01)^2 = 1.0201

(1.01)^3 = 1.030301

(1.01)^4 = 1.04060401

(1.01)^5 = 1.05101005

(1.01)^6 = 1.06152015

(1.01)^7 = 1.07213535

(1.01)^8 = 1.08285670

(1.01)^9 = 1.09368527

(1.01)^10 ≈ 1.10462210

Yes, so (1.01)^10 ≈ 1.10462210

Then, (1.01)^20 = [(1.01)^10]^2 ≈ (1.10462210)^2

Let me compute that:

1.10462210 * 1.10462210

First, compute 1.1 * 1.1 = 1.21

But more accurately:

Let me write 1.10462210 * 1.10462210.

Let me use the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 1.1, b = 0.00462210

Wait, maybe that's not helpful. Alternatively, use decimal multiplication.

Alternatively, approximate:

Let me note that 1.10462210^2 ≈ ?

Well, 1.1^2 = 1.21

But since it's 1.1046..., which is a bit more than 1.1, so the square will be a bit more than 1.21.

Alternatively, use the value of (1.10462210)^2:

Let me compute 1.1046221 * 1.1046221:

First, multiply 1.1046221 * 1.1046221:

Let me use the fact that (1 + 0.1046221)^2 = 1 + 2*0.1046221 + (0.1046221)^2 ≈ 1 + 0.2092442 + 0.010946 ≈ 1.22019. Wait, but that seems conflicting with previous thought. Wait, no, because 1.1046221 is 1 + 0.1046221, so squaring that gives 1 + 2*0.1046221 + (0.1046221)^2 ≈ 1 + 0.2092442 + 0.010946 ≈ 1.22019. However, if I compute 1.1046221 * 1.1046221 directly:

Let me do it step by step:

First, 1 * 1.1046221 = 1.1046221

Then, 0.1 * 1.1046221 = 0.11046221

Then, 0.0046221 * 1.1046221 ≈ ?

Wait, this is getting too complicated. Maybe I should use a better approach. Alternatively, since I know that (1.01)^10 ≈ 1.104622, then (1.01)^20 ≈ (1.104622)^2. Let me use a calculator-like approach here. Let me approximate 1.104622 * 1.104622.

Let me write 1.104622 * 1.104622:

First, multiply 1.104622 * 1 = 1.104622

Then, 1.104622 * 0.1 = 0.1104622

Then, 1.104622 * 0.004 = 0.004418488

Then, 1.104622 * 0.0006 = 0.000662773

Then, 1.104622 * 0.00002 = 0.000022092

Then, 1.104622 * 0.000002 = 0.000002209

Adding all these up:

Start with 1.104622

Add 0.1104622: 1.104622 + 0.1104622 = 1.2150842

Add 0.004418488: 1.2150842 + 0.004418488 ≈ 1.219502688

Add 0.000662773: ≈ 1.219502688 + 0.000662773 ≈ 1.220165461

Add 0.000022092: ≈ 1.220165461 + 0.000022092 ≈ 1.220187553

Add 0.000002209: ≈ 1.220187553 + 0.000002209 ≈ 1.220189762

So, approximately 1.22019. Therefore, (1.01)^20 ≈ 1.22019.

Then, (1.01)^30 = (1.01)^20 * (1.01)^10 ≈ 1.22019 * 1.104622.

Let me compute that. 1.22019 * 1.104622.

Again, breaking it down:

First, 1 * 1.104622 = 1.104622

Then, 0.2 * 1.104622 = 0.2209244

Then, 0.02 * 1.104622 = 0.02209244

Then, 0.00019 * 1.104622 ≈ 0.000209878

Adding them up:

Start with 1.104622 + 0.2209244 = 1.3255464

Add 0.02209244: 1.3255464 + 0.02209244 ≈ 1.34763884

Add 0.000209878: ≈ 1.34763884 + 0.000209878 ≈ 1.3478487

Therefore, (1.01)^30 ≈ 1.3478487. So, approximately 1.34785. Therefore, my initial approximation of 1.3478 was very close. Therefore, multiplying by 24,000 gives:

24,000 * 1.34785 ≈ ?

Earlier, I had approximated this as 32,347.2, but let me compute it more accurately.

First, 24,000 * 1.34785.

Let me write 1.34785 as 1 + 0.3 + 0.04 + 0.007 + 0.00085.

But maybe breaking it down differently:

Let me compute 24,000 * 1.34785.

First, note that 24,000 * 1.34785 = 24,000 * (1 + 0.3 + 0.04 + 0.007 + 0.00085) = 24,000 + 24,000*0.3 + 24,000*0.04 + 24,000*0.007 + 24,000*0.00085.

Compute each term:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.007 = 168

24,000 * 0.00085 = 24,000 * 0.0008 + 24,000 * 0.00005 = 19.2 + 1.2 = 20.4

Now, adding all these up:

Start with 24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 168 = 32,328

32,328 + 20.4 = 32,348.4

Therefore, approximately $32,348.40. However, this is based on the approximation of (1.01)^30 ≈ 1.34785. However, to be more precise, maybe I should use a calculator value for (1.01)^30?

Alternatively, perhaps I can use the formula for compound interest with more accurate decimal places. Alternatively, maybe I can use logarithms or another method. However, since I don't have a calculator here, but given that my step-by-step multiplication gave me approximately 1.34785, which led to 32,348.4, which is approximately $32,348.40. However, since the question says to round to the nearest whole number, so if the exact value is slightly more or less than this, it might round up or down.

But perhaps my approximation of (1.01)^30 is slightly off. Let me check with another method. Let me use the formula for compound interest with more precise calculation.

Alternatively, I can use the Taylor series expansion for (1.01)^30. But that might be too time-consuming. Alternatively, I can use semi-annual calculations. Wait, but I already did the step-by-step calculation for (1.01)^10, then squared it to get (1.01)^20, then multiplied by (1.01)^10 to get (1.01)^30, and that gave me approximately 1.34785. Let me check with another approach.

Alternatively, I can use the rule that (1 + r)^n ≈ e^{nr} when r is small. However, here r is 0.01, n is 30, so nr = 0.3. Then, e^{0.3} ≈ 1.349858. Which is very close to our previous value of 1.34785. Wait, but why is there a discrepancy?

Because the approximation (1 + r)^n ≈ e^{nr} is only accurate when r is very small and n is large? Wait, no, actually, the approximation (1 + r/n)^{nt} ≈ e^{rt} is the standard one. But here, we have (1 + r)^n where r is 0.01 and n is 30. So, it's not the same as the standard approximation. Therefore, the difference between e^{0.3} ≈ 1.349858 and our calculated value of approximately 1.34785 is about 0.002, which is about 0.15% difference. So, that's a small difference. Therefore, if I use e^{0.3} ≈ 1.349858, then multiplying by 24,000 gives 24,000 * 1.349858 ≈ ?

Let me compute that. 24,000 * 1.349858.

Again, breaking it down:

24,000 * 1.349858 = 24,000 * (1 + 0.3 + 0.04 + 0.009858)

Wait, 1.349858 is 1 + 0.3 + 0.04 + 0.009858. Let me compute:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.009858 ≈ ?

First, 24,000 * 0.01 = 240

So, 24,000 * 0.009858 = 24,000 * (0.01 - 0.000142) = 240 - 24,000 * 0.000142

Compute 24,000 * 0.000142 = 24,000 * 0.0001 + 24,000 * 0.000042 = 2.4 + 1.008 = 3.408

Therefore, 24,000 * 0.009858 ≈ 240 - 3.408 = 236.592

Therefore, total amount is:

24,000 + 7,200 + 960 + 236.592 = ?

24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 236.592 ≈ 32,396.592

So, approximately $32,396.59. However, this is using the e^{0.3} approximation, which is higher than our previous value of 1.34785. Therefore, there is a discrepancy here. Which one is more accurate?

Earlier, when I calculated (1.01)^30 step by step, I got approximately 1.34785, but using the e^{0.3} approximation, I get 1.349858. Which is correct?

Well, let me check with another method. Let me compute (1.01)^30 using semi-annual calculations. Wait, but I already did that. Alternatively, maybe I can use the formula for compound interest with more precise decimal places.

Alternatively, perhaps I can use the fact that (1.01)^30 is equal to e^{30 * ln(1.01)}. Let me compute ln(1.01) more accurately. Let me recall that ln(1.01) can be calculated using the Taylor series:

ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - ... for |x| < 1.

Here, x = 0.01, so:

ln(1.01) = 0.01 - (0.01)^2 / 2 + (0.01)^3 / 3 - (0.01)^4 / 4 + (0.01)^5 / 5 - ...

Compute up to several terms:

First term: 0.01

Second term: - (0.0001)/2 = -0.00005

Third term: + (0.000001)/3 ≈ +0.0000003333

Fourth term: - (0.00000001)/4 ≈ -0.0000000025

Fifth term: + (0.0000000001)/5 ≈ +0.00000000002

So, adding these up:

0.01 - 0.00005 = 0.00995

Plus 0.0000003333 ≈ 0.0099503333

Minus 0.0000000025 ≈ 0.0099503308

Plus negligible terms. So, ln(1.01) ≈ 0.0099503308.

Therefore, multiplying by 30 gives 30 * 0.0099503308 ≈ 0.298509924.

Now, compute e^{0.298509924}.

We can use the Taylor series expansion for e^x around x = 0.3, but maybe it's easier to use known values. Let me recall that e^0.298509924.

We know that e^0.3 ≈ 1.3498588075760032.

But our exponent is 0.298509924, which is 0.3 - 0.001490076.

Therefore, e^{0.298509924} = e^{0.3 - 0.001490076} = e^{0.3} * e^{-0.001490076}.

We know e^{0.3} ≈ 1.3498588075760032.

Now, compute e^{-0.001490076} ≈ 1 - 0.001490076 + (0.001490076)^2 / 2 - ... using the Taylor series for e^x around x=0.

So, e^{-0.001490076} ≈ 1 - 0.001490076 + (0.001490076)^2 / 2.

Compute:

First term: 1

Second term: -0.001490076

Third term: (0.001490076)^2 / 2 ≈ (0.0000022202) / 2 ≈ 0.0000011101

Therefore, total ≈ 1 - 0.001490076 + 0.0000011101 ≈ 0.998511034.

Therefore, e^{0.298509924} ≈ 1.3498588075760032 * 0.998511034 ≈ ?

Let me compute that:

First, multiply 1.3498588075760032 * 0.998511034.

Let me approximate this as follows:

Let me write 1.3498588 * 0.998511034.

Let me note that 1.3498588 * 0.998511034 ≈ 1.3498588 * (1 - 0.001488966) ≈ 1.3498588 - 1.3498588 * 0.001488966.

Compute 1.3498588 * 0.001488966 ≈ ?

First, approximate 1.3498588 * 0.001 = 0.0013498588

Then, 1.3498588 * 0.000488966 ≈ ?

Compute 1.3498588 * 0.0004 = 0.0005399435

1.3498588 * 0.000088966 ≈ approx 1.3498588 * 0.00008 = 0.0001079887 + 1.3498588 * 0.000008966 ≈ approx 0.0000121.

So total approx 0.0001079887 + 0.0000121 ≈ 0.0001200887.

Therefore, total of 0.0005399435 + 0.0001200887 ≈ 0.0006600322.

Therefore, total of 0.0013498588 + 0.0006600322 ≈ 0.002009891.

Therefore, the entire expression is approx 1.3498588 - 0.002009891 ≈ 1.3478489.

Therefore, e^{0.298509924} ≈ 1.3478489. Which matches our previous step-by-step calculation of (1.01)^30 ≈ 1.34785. Therefore, this is accurate. Therefore, the exact value of (1.01)^30 is approximately 1.3478489.

Therefore, multiplying by 24,000:

24,000 * 1.3478489 ≈ ?

As calculated earlier, breaking it down:

24,000 * 1.3478489 ≈ 24,000 * 1.34785 ≈ 32,348.4. However, let me do a more precise calculation.

Let me note that 1.3478489 = 1.3478 + 0.0000489.

So, 24,000 * 1.3478 = 32,347.2 (as calculated earlier)

Then, 24,000 * 0.0000489 = 24,000 * 0.00004 + 24,000 * 0.0000089 = 0.96 + 0.2136 = 1.1736

Therefore, total amount ≈ 32,347.2 + 1.1736 ≈ 32,348.3736.

Therefore, approximately $32,348.37. So, about $32,348.37. Therefore, rounding to the nearest whole number, it would be $32,348.

But wait, let me check if there is any more precise calculation needed. Because depending on the exact value of (1.01)^30, the amount could be slightly more or less. However, given that we have calculated (1.01)^30 ≈ 1.3478489, which is approximately 1.347849, so multiplying by 24,000:

Let me compute 24,000 * 1.347849.

First, note that 24,000 * 1.347849 = 24,000 * 1 + 24,000 * 0.3 + 24,000 * 0.04 + 24,000 * 0.007 + 24,000 * 0.000849.

Wait, breaking down 1.347849 as 1 + 0.3 + 0.04 + 0.007 + 0.000849.

So:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.007 = 168

24,000 * 0.000849 = ?

Compute 24,000 * 0.0008 = 19.2

24,000 * 0.000049 = 24,000 * 0.00004 + 24,000 * 0.000009 = 0.96 + 0.216 = 1.176

Therefore, total for 0.000849 is 19.2 + 1.176 = 20.376

Therefore, adding all together:

24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 168 = 32,328

32,328 + 20.376 = 32,348.376

Therefore, approximately $32,348.38. So, $32,348.38, which rounds to $32,348 when rounded to the nearest whole number.

But wait, let me check if there is a more accurate way. Maybe I can use more decimal places in the exponent?

Alternatively, perhaps I can use the exact value of (1.01)^30. However, without a calculator, it's difficult. However, given that we have calculated it as approximately 1.3478489, which gives us approximately $32,348.38, which is about $32,348.38. Therefore, the nearest whole number is $32,348.

But let me check with another approach. Let me use semi-annual calculations. Wait, but we already did that. Alternatively, maybe I can use the formula for compound interest with more precise decimal places.

Alternatively, perhaps I can use the fact that (1.01)^30 is equal to 1.3478489, so multiplying by 24,000 gives 24,000 * 1.3478489. Let me compute this multiplication more accurately.

Let me write 24,000 * 1.3478489.

First, note that 24,000 * 1.3478489 = 24,000 * 1.3478489.

Let me convert 1.3478489 into a decimal and multiply:

Let me write 1.3478489 as 1.3478489.

Multiplying by 24,000:

First, multiply 1.3478489 by 24,000.

Note that 1.3478489 * 24,000 = 1.3478489 * 24 * 1,000.

Compute 1.3478489 * 24 first.

Compute 1.3478489 * 20 = 26.956978

Compute 1.3478489 * 4 = 5.3913956

Add them together: 26.956978 + 5.3913956 = 32.3483736

Therefore, 1.3478489 * 24 = 32.3483736

Then, multiply by 1,000: 32.3483736 * 1,000 = 32,348.3736

Therefore, the exact value is approximately $32,348.3736, which is approximately $32,348.37 when rounded to the nearest cent. Therefore, when rounded to the nearest whole number, it is $32,348.

But wait, let me check if there is a possibility that the exact value is slightly more than $32,348.50, which would round up to $32,349. However, given that the decimal part is 0.37, which is less than 0.5, it should round down to $32,348.

But let me check once again if my calculation of (1.01)^30 is correct. Because if there is an error in that, then the final answer would be wrong.

Earlier, we calculated (1.01)^10 ≈ 1.1046221, then (1.01)^20 ≈ 1.22019, then (1.01)^30 ≈ 1.22019 * 1.104622 ≈ 1.34785. Let me verify this multiplication again.

Compute 1.22019 * 1.104622:

Let me do this multiplication more carefully.

First, write 1.22019 * 1.104622.

Let me use the standard multiplication method:

Multiply 1.22019 by 1.104622.

First, ignore the decimals and consider them as integers:

But this might be too time-consuming. Alternatively, use the following approach:

Let me approximate 1.22019 * 1.104622.

Let me note that 1.22019 * 1.104622 ≈ 1.22 * 1.1046 ≈ ?

Compute 1.22 * 1.1046:

First, 1 * 1.1046 = 1.1046

0.2 * 1.1046 = 0.22092

0.02 * 1.1046 = 0.022092

Adding them up: 1.1046 + 0.22092 = 1.32552 + 0.022092 = 1.347612

Therefore, approx 1.3476, which is very close to our previous value of 1.34785. Therefore, the exact value is approximately 1.3478, which gives us the total amount as approximately $32,348.37. Therefore, the answer should be $32,348 when rounded to the nearest whole number.

However, to be absolutely sure, let me check with another method. Let me use the formula for compound interest with more precise exponentiation.

Alternatively, I can use the fact that (1.01)^30 can be calculated as follows:

We can use the formula (1 + r)^n = e^{n * ln(1 + r)}.

We already calculated ln(1.01) ≈ 0.00995033, so n * ln(1 + r) = 30 * 0.00995033 ≈ 0.2985099.

Then, e^{0.2985099} ≈ 1.3478489 as calculated before. Therefore, multiplying by 24,000 gives 32,348.37, which rounds to 32,348.

But let me check with another resource or formula. Wait, since I can't actually look up a calculator, but maybe I can use semi-annual calculations step by step for 30 periods. However, that would take a long time, but let me try for a few periods to see if the value is consistent.

For example, let me compute (1.01)^10 ≈ 1.104622 as before.

Then, (1.01)^15 = (1.01)^10 * (1.01)^5 ≈ 1.104622 * 1.051010 ≈ ?

Compute 1.104622 * 1.051010:

First, 1 * 1.051010 = 1.051010

0.1 * 1.051010 = 0.105101

0.004622 * 1.051010 ≈ 0.004855

Adding them up: 1.051010 + 0.105101 = 1.156111 + 0.004855 ≈ 1.160966. Therefore, (1.01)^15 ≈ 1.160966.

Then, (1.01)^20 = (1.01)^15 * (1.01)^5 ≈ 1.160966 * 1.051010 ≈ ?

Compute 1.160966 * 1.051010:

First, 1 * 1.051010 = 1.051010

0.1 * 1.051010 = 0.105101

0.06 * 1.051010 = 0.0630606

0.000966 * 1.051010 ≈ 0.001014

Adding them up:

1.051010 + 0.105101 = 1.156111 + 0.0630606 = 1.2191716 + 0.001014 ≈ 1.2201856. Which matches our previous value of (1.01)^20 ≈ 1.22019. Therefore, that seems correct.

Then, (1.01)^30 = (1.01)^20 * (1.01)^10 ≈ 1.2201856 * 1.104622 ≈ ?

Compute 1.2201856 * 1.104622:

Again, breaking it down:

First, 1 * 1.104622 = 1.104622

0.2 * 1.104622 = 0.2209244

0.02 * 1.104622 = 0.02209244

0.0001856 * 1.104622 ≈ 0.000205

Adding them up:

1.104622 + 0.2209244 = 1.3255464 + 0.02209244 = 1.34763884 + 0.000205 ≈ 1.34784384. Which is approximately 1.347844, which matches our previous value of 1.3478489. Therefore, this confirms that (1.01)^30 ≈ 1.347844.

Therefore, multiplying by 24,000 gives 24,000 * 1.347844 ≈ 32,348.256, which is approximately $32,348.26, which rounds to $32,348.

Therefore, after all these calculations and cross-verifying, I can be confident that the total amount after five years is approximately $32,348 when rounded to the nearest whole number.

However, just to be absolutely thorough, let me check if there's any chance that the interest is compounded differently. The problem says "bi-monthly interest", which means every two months. So, that is, the interest is compounded every two months, which is what I considered. So, the formula A = P*(1 + r)^t is correct here, where r = 0.01, t = 30.

Alternatively, if the problem had said that the annual interest rate is 1% compounded bi-monthly, that would be different. But the problem says "1% bi-monthly interest", meaning that every two months, the investment increases by 1%. Therefore, the rate per period is 1%, and the number of periods is 30. Therefore, the calculation is correct.

Therefore, I think my answer is correct. So, the total amount after five years is approximately $32,348.

But just to make sure, let me check with another approach. Let me use the formula A = P * e^{rt} if it were continuous compounding, but that's not the case here. However, just for comparison, if it were continuous compounding, the amount would be 24,000 * e^{0.01*30} = 24,000 * e^{0.3} ≈ 24,000 * 1.349858 ≈ 32,396.59, which is higher than our value. But since it's compounded discretely every two months, the amount should be less than continuous compounding, which matches our result of ~$32,348, which is less than ~$32,396. So that seems reasonable.

Alternatively, if it were compounded monthly, the amount would be higher than bi-monthly. Wait, no, if it's compounded more frequently, the amount would be higher. Since bi-monthly is every two months, which is less frequent than monthly. Therefore, the amount should be less than if it were compounded monthly. However, since we are compounding every two months, which is less frequent than monthly, so the amount should be less than monthly compounding. However, since we don't have the monthly compounding value, it's hard to compare. But given that our value is ~$32,348, which is less than the continuous compounding value, which is reasonable.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{32348}

To determine the total value of a $24,000 investment after five years with **1% bi-monthly interest**, we need to apply the **compound interest formula**:

$$
A = P(1 + r)^t
$$

Where:
- $ A $ is the final amount,
- $ P = 24,000 $ is the principal,
- $ r = 0.01 $ is the interest rate per period (1%),
- $ t = 30 $ is the number of compounding periods.

---

### Step 1: Determine the Number of Periods

Since the interest is compounded **bi-monthly**, that means it is compounded **every two months**. Over five years:

$$
\text{Number of periods} = \frac{5 \text{ years} \times 12 \text{ months/year}}{2 \text{ months/period}} = 30 \text{ periods}
$$

---

### Step 2: Apply the Compound Interest Formula

$$
A = 24,000 \times (1 + 0.01)^{30} = 24,000 \times (1.01)^{30}
$$

We now compute $ (1.01)^{30} $. Using precise calculation or a calculator:

$$
(1.01)^{30} \approx 1.3478489
$$

---

### Step 3: Multiply by the Principal

$$
A = 24,000 \times 1.3478489 \approx 32,348.37
$$

---

### Step 4: Round to the Nearest Whole Number

$$
\text{Final amount} \approx \boxed{32348}
$$

---

### Final Answer

$$
\boxed{32348}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how much money will be in the investment after five years if it's compounded bi-monthly at 1% interest. Let me start by understanding the problem step by step.

First, the initial investment is $24,000. The interest is 1% every two months. That means every two months, the investment increases by 1% of its current value. So, this is compound interest, right? Because each time, the interest is calculated on the current amount, not just the original principal.

Now, the question is asking for the total amount after five years. So, I need to figure out how many bi-monthly periods there are in five years. Let me recall that "bi-monthly" means every two months. So, how many two-month periods are in a year? Well, a year has 12 months, so 12 divided by 2 is 6. Therefore, there are 6 bi-monthly periods in a year. Therefore, over five years, that would be 5 times 6, which is 30 periods. Let me check that again. If each period is two months, then 5 years is 60 months. Divided by 2, that's 30 periods. Yep, that seems right.

So, the formula for compound interest is A = P(1 + r/n)^(nt), but wait, in this case, the compounding is happening every two months, so maybe I need to adjust the formula accordingly. Wait, actually, since the interest is compounded every two months, the formula would be A = P(1 + r)^t, where r is the interest rate per period, and t is the number of periods. Let me confirm.

In standard compound interest, A = P(1 + r/n)^(nt), where n is the number of times compounded per year. But here, the compounding is every two months, so if we consider the rate per period, it's 1% per period, and the number of periods is 30. So, maybe it's simpler. Let me think.

Alternatively, since the interest is 1% every two months, then each period, the amount is multiplied by 1.01. Therefore, after t periods, the amount is P*(1.01)^t. So, yes, that seems straightforward. So, in this case, P is 24,000, t is 30, and the rate per period is 1%, so 0.01. Therefore, A = 24000*(1.01)^30. Then, compute that value and round to the nearest whole number.

But let me make sure I didn't make a mistake here. Let me check the formula again. If it's compounded every two months, then the formula is indeed A = P*(1 + r)^t, where r is the rate per period, and t is the number of periods. So, since each period is two months, and the rate is 1% per period, then yes, that formula applies. So, that seems correct.

Alternatively, if I wanted to use the standard formula, I could think of it as annual compounding, but since it's bi-monthly, maybe not. Let me think. If I wanted to use the standard formula, I would need to know the annual interest rate. But here, the problem states that the interest is 1% bi-monthly. So, that is, every two months, the interest is 1%. Therefore, the rate per period is 1%, and the number of periods is 30. Therefore, the formula is correct.

So, moving forward with that. Let me compute 24000*(1.01)^30. Now, I need to calculate (1.01)^30. Let me recall that (1 + r)^t can be calculated using logarithms or using the formula for compound interest. Alternatively, I can use the rule of 72 to estimate, but since I need an exact value, I need to calculate it more precisely.

Alternatively, I can use the formula for compound interest. Let me try to compute (1.01)^30. Let me remember that ln(1.01) is approximately 0.00995, so multiplying by 30 gives approximately 0.2985. Then, exponentiating gives e^0.2985 ≈ 1.347. But this is an approximation. However, since I need an exact value, maybe I should use a calculator? Wait, but since this is a problem-solving scenario, perhaps I need to compute it step by step or use a better approximation.

Alternatively, maybe I can use the binomial theorem? But that might be too tedious. Alternatively, I can use the fact that (1.01)^30 is equal to e^(30*ln(1.01)). Let me compute ln(1.01). Let me recall that ln(1.01) is approximately 0.00995033. Let me check with more precision. Let me remember that ln(1+x) ≈ x - x^2/2 + x^3/3 - x^4/4 + ... for small x. Here, x is 0.01, so ln(1.01) ≈ 0.01 - 0.00005 + 0.00000333 - ... which is approximately 0.00995033. So, multiplying by 30 gives 0.2985099. Then, e^0.2985099. Let me compute e^0.2985.

We know that e^0.3 is approximately 1.349858. Since 0.2985 is slightly less than 0.3, so e^0.2985 ≈ e^(0.3 - 0.0015) = e^0.3 * e^(-0.0015). We know e^(-0.0015) ≈ 1 - 0.0015 + ... ≈ 0.9985. Therefore, e^0.2985 ≈ 1.349858 * 0.9985 ≈ 1.349858 - 1.349858*0.0015 ≈ 1.349858 - 0.002024787 ≈ 1.34783. Therefore, approximately 1.3478. Therefore, (1.01)^30 ≈ 1.3478. Therefore, multiplying by 24,000 gives 24,000 * 1.3478 ≈ ?

Let me compute 24,000 * 1.3478. Let me break it down:

First, 24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.0078 = ?

Wait, 1.3478 is 1 + 0.3 + 0.04 + 0.0078. Let me check:

1 + 0.3 = 1.3

1.3 + 0.04 = 1.34

1.34 + 0.0078 = 1.3478. Yes.

So, 24,000 * 1.3478 = 24,000 * 1 + 24,000 * 0.3 + 24,000 * 0.04 + 24,000 * 0.0078.

So, 24,000 + 7,200 + 960 + ?

Compute 24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

Now, 24,000 * 0.0078. Let me compute 24,000 * 0.0078.

First, 24,000 * 0.007 = 168

24,000 * 0.0008 = 19.2

So, total is 168 + 19.2 = 187.2

Therefore, total amount is 32,160 + 187.2 = 32,347.2

So, approximately $32,347.20. But this is based on my approximation of (1.01)^30 ≈ 1.3478. However, I need to check if this approximation is accurate enough. Because if my estimation of (1.01)^30 is off, then my final answer will be off.

Alternatively, maybe I can calculate (1.01)^30 more accurately. Let me try to compute it step by step. Let me recall that (1.01)^10 is approximately 1.104622, and then (1.104622)^3 would be (1.01)^30. Let me check that.

First, compute (1.01)^10. Let me compute that:

(1.01)^1 = 1.01

(1.01)^2 = 1.0201

(1.01)^3 = 1.030301

(1.01)^4 = 1.04060401

(1.01)^5 = 1.05101005

(1.01)^6 = 1.06152015

(1.01)^7 = 1.07213535

(1.01)^8 = 1.08285670

(1.01)^9 = 1.09368527

(1.01)^10 ≈ 1.10462210

Yes, so (1.01)^10 ≈ 1.10462210

Then, (1.01)^20 = [(1.01)^10]^2 ≈ (1.10462210)^2

Let me compute that:

1.10462210 * 1.10462210

First, compute 1.1 * 1.1 = 1.21

But more accurately:

Let me write 1.10462210 * 1.10462210.

Let me use the formula (a + b)^2 = a^2 + 2ab + b^2, where a = 1.1, b = 0.00462210

Wait, maybe that's not helpful. Alternatively, use decimal multiplication.

Alternatively, approximate:

Let me note that 1.10462210^2 ≈ ?

Well, 1.1^2 = 1.21

But since it's 1.1046..., which is a bit more than 1.1, so the square will be a bit more than 1.21.

Alternatively, use the value of (1.10462210)^2:

Let me compute 1.1046221 * 1.1046221:

First, multiply 1.1046221 * 1.1046221:

Let me use the fact that (1 + 0.1046221)^2 = 1 + 2*0.1046221 + (0.1046221)^2 ≈ 1 + 0.2092442 + 0.010946 ≈ 1.22019. Wait, but that seems conflicting with previous thought. Wait, no, because 1.1046221 is 1 + 0.1046221, so squaring that gives 1 + 2*0.1046221 + (0.1046221)^2 ≈ 1 + 0.2092442 + 0.010946 ≈ 1.22019. However, if I compute 1.1046221 * 1.1046221 directly:

Let me do it step by step:

First, 1 * 1.1046221 = 1.1046221

Then, 0.1 * 1.1046221 = 0.11046221

Then, 0.0046221 * 1.1046221 ≈ ?

Wait, this is getting too complicated. Maybe I should use a better approach. Alternatively, since I know that (1.01)^10 ≈ 1.104622, then (1.01)^20 ≈ (1.104622)^2. Let me use a calculator-like approach here. Let me approximate 1.104622 * 1.104622.

Let me write 1.104622 * 1.104622:

First, multiply 1.104622 * 1 = 1.104622

Then, 1.104622 * 0.1 = 0.1104622

Then, 1.104622 * 0.004 = 0.004418488

Then, 1.104622 * 0.0006 = 0.000662773

Then, 1.104622 * 0.00002 = 0.000022092

Then, 1.104622 * 0.000002 = 0.000002209

Adding all these up:

Start with 1.104622

Add 0.1104622: 1.104622 + 0.1104622 = 1.2150842

Add 0.004418488: 1.2150842 + 0.004418488 ≈ 1.219502688

Add 0.000662773: ≈ 1.219502688 + 0.000662773 ≈ 1.220165461

Add 0.000022092: ≈ 1.220165461 + 0.000022092 ≈ 1.220187553

Add 0.000002209: ≈ 1.220187553 + 0.000002209 ≈ 1.220189762

So, approximately 1.22019. Therefore, (1.01)^20 ≈ 1.22019.

Then, (1.01)^30 = (1.01)^20 * (1.01)^10 ≈ 1.22019 * 1.104622.

Let me compute that. 1.22019 * 1.104622.

Again, breaking it down:

First, 1 * 1.104622 = 1.104622

Then, 0.2 * 1.104622 = 0.2209244

Then, 0.02 * 1.104622 = 0.02209244

Then, 0.00019 * 1.104622 ≈ 0.000209878

Adding them up:

Start with 1.104622 + 0.2209244 = 1.3255464

Add 0.02209244: 1.3255464 + 0.02209244 ≈ 1.34763884

Add 0.000209878: ≈ 1.34763884 + 0.000209878 ≈ 1.3478487

Therefore, (1.01)^30 ≈ 1.3478487. So, approximately 1.34785. Therefore, my initial approximation of 1.3478 was very close. Therefore, multiplying by 24,000 gives:

24,000 * 1.34785 ≈ ?

Earlier, I had approximated this as 32,347.2, but let me compute it more accurately.

First, 24,000 * 1.34785.

Let me write 1.34785 as 1 + 0.3 + 0.04 + 0.007 + 0.00085.

But maybe breaking it down differently:

Let me compute 24,000 * 1.34785.

First, note that 24,000 * 1.34785 = 24,000 * (1 + 0.3 + 0.04 + 0.007 + 0.00085) = 24,000 + 24,000*0.3 + 24,000*0.04 + 24,000*0.007 + 24,000*0.00085.

Compute each term:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.007 = 168

24,000 * 0.00085 = 24,000 * 0.0008 + 24,000 * 0.00005 = 19.2 + 1.2 = 20.4

Now, adding all these up:

Start with 24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 168 = 32,328

32,328 + 20.4 = 32,348.4

Therefore, approximately $32,348.40. However, this is based on the approximation of (1.01)^30 ≈ 1.34785. However, to be more precise, maybe I should use a calculator value for (1.01)^30?

Alternatively, perhaps I can use the formula for compound interest with more accurate decimal places. Alternatively, maybe I can use logarithms or another method. However, since I don't have a calculator here, but given that my step-by-step multiplication gave me approximately 1.34785, which led to 32,348.4, which is approximately $32,348.40. However, since the question says to round to the nearest whole number, so if the exact value is slightly more or less than this, it might round up or down.

But perhaps my approximation of (1.01)^30 is slightly off. Let me check with another method. Let me use the formula for compound interest with more precise calculation.

Alternatively, I can use the Taylor series expansion for (1.01)^30. But that might be too time-consuming. Alternatively, I can use semi-annual calculations. Wait, but I already did the step-by-step calculation for (1.01)^10, then squared it to get (1.01)^20, then multiplied by (1.01)^10 to get (1.01)^30, and that gave me approximately 1.34785. Let me check with another approach.

Alternatively, I can use the rule that (1 + r)^n ≈ e^{nr} when r is small. However, here r is 0.01, n is 30, so nr = 0.3. Then, e^{0.3} ≈ 1.349858. Which is very close to our previous value of 1.34785. Wait, but why is there a discrepancy?

Because the approximation (1 + r)^n ≈ e^{nr} is only accurate when r is very small and n is large? Wait, no, actually, the approximation (1 + r/n)^{nt} ≈ e^{rt} is the standard one. But here, we have (1 + r)^n where r is 0.01 and n is 30. So, it's not the same as the standard approximation. Therefore, the difference between e^{0.3} ≈ 1.349858 and our calculated value of approximately 1.34785 is about 0.002, which is about 0.15% difference. So, that's a small difference. Therefore, if I use e^{0.3} ≈ 1.349858, then multiplying by 24,000 gives 24,000 * 1.349858 ≈ ?

Let me compute that. 24,000 * 1.349858.

Again, breaking it down:

24,000 * 1.349858 = 24,000 * (1 + 0.3 + 0.04 + 0.009858)

Wait, 1.349858 is 1 + 0.3 + 0.04 + 0.009858. Let me compute:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.009858 ≈ ?

First, 24,000 * 0.01 = 240

So, 24,000 * 0.009858 = 24,000 * (0.01 - 0.000142) = 240 - 24,000 * 0.000142

Compute 24,000 * 0.000142 = 24,000 * 0.0001 + 24,000 * 0.000042 = 2.4 + 1.008 = 3.408

Therefore, 24,000 * 0.009858 ≈ 240 - 3.408 = 236.592

Therefore, total amount is:

24,000 + 7,200 + 960 + 236.592 = ?

24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 236.592 ≈ 32,396.592

So, approximately $32,396.59. However, this is using the e^{0.3} approximation, which is higher than our previous value of 1.34785. Therefore, there is a discrepancy here. Which one is more accurate?

Earlier, when I calculated (1.01)^30 step by step, I got approximately 1.34785, but using the e^{0.3} approximation, I get 1.349858. Which is correct?

Well, let me check with another method. Let me compute (1.01)^30 using semi-annual calculations. Wait, but I already did that. Alternatively, maybe I can use the formula for compound interest with more precise decimal places.

Alternatively, perhaps I can use the fact that (1.01)^30 is equal to e^{30 * ln(1.01)}. Let me compute ln(1.01) more accurately. Let me recall that ln(1.01) can be calculated using the Taylor series:

ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - ... for |x| < 1.

Here, x = 0.01, so:

ln(1.01) = 0.01 - (0.01)^2 / 2 + (0.01)^3 / 3 - (0.01)^4 / 4 + (0.01)^5 / 5 - ...

Compute up to several terms:

First term: 0.01

Second term: - (0.0001)/2 = -0.00005

Third term: + (0.000001)/3 ≈ +0.0000003333

Fourth term: - (0.00000001)/4 ≈ -0.0000000025

Fifth term: + (0.0000000001)/5 ≈ +0.00000000002

So, adding these up:

0.01 - 0.00005 = 0.00995

Plus 0.0000003333 ≈ 0.0099503333

Minus 0.0000000025 ≈ 0.0099503308

Plus negligible terms. So, ln(1.01) ≈ 0.0099503308.

Therefore, multiplying by 30 gives 30 * 0.0099503308 ≈ 0.298509924.

Now, compute e^{0.298509924}.

We can use the Taylor series expansion for e^x around x = 0.3, but maybe it's easier to use known values. Let me recall that e^0.298509924.

We know that e^0.3 ≈ 1.3498588075760032.

But our exponent is 0.298509924, which is 0.3 - 0.001490076.

Therefore, e^{0.298509924} = e^{0.3 - 0.001490076} = e^{0.3} * e^{-0.001490076}.

We know e^{0.3} ≈ 1.3498588075760032.

Now, compute e^{-0.001490076} ≈ 1 - 0.001490076 + (0.001490076)^2 / 2 - ... using the Taylor series for e^x around x=0.

So, e^{-0.001490076} ≈ 1 - 0.001490076 + (0.001490076)^2 / 2.

Compute:

First term: 1

Second term: -0.001490076

Third term: (0.001490076)^2 / 2 ≈ (0.0000022202) / 2 ≈ 0.0000011101

Therefore, total ≈ 1 - 0.001490076 + 0.0000011101 ≈ 0.998511034.

Therefore, e^{0.298509924} ≈ 1.3498588075760032 * 0.998511034 ≈ ?

Let me compute that:

First, multiply 1.3498588075760032 * 0.998511034.

Let me approximate this as follows:

Let me write 1.3498588 * 0.998511034.

Let me note that 1.3498588 * 0.998511034 ≈ 1.3498588 * (1 - 0.001488966) ≈ 1.3498588 - 1.3498588 * 0.001488966.

Compute 1.3498588 * 0.001488966 ≈ ?

First, approximate 1.3498588 * 0.001 = 0.0013498588

Then, 1.3498588 * 0.000488966 ≈ ?

Compute 1.3498588 * 0.0004 = 0.0005399435

1.3498588 * 0.000088966 ≈ approx 1.3498588 * 0.00008 = 0.0001079887 + 1.3498588 * 0.000008966 ≈ approx 0.0000121.

So total approx 0.0001079887 + 0.0000121 ≈ 0.0001200887.

Therefore, total of 0.0005399435 + 0.0001200887 ≈ 0.0006600322.

Therefore, total of 0.0013498588 + 0.0006600322 ≈ 0.002009891.

Therefore, the entire expression is approx 1.3498588 - 0.002009891 ≈ 1.3478489.

Therefore, e^{0.298509924} ≈ 1.3478489. Which matches our previous step-by-step calculation of (1.01)^30 ≈ 1.34785. Therefore, this is accurate. Therefore, the exact value of (1.01)^30 is approximately 1.3478489.

Therefore, multiplying by 24,000:

24,000 * 1.3478489 ≈ ?

As calculated earlier, breaking it down:

24,000 * 1.3478489 ≈ 24,000 * 1.34785 ≈ 32,348.4. However, let me do a more precise calculation.

Let me note that 1.3478489 = 1.3478 + 0.0000489.

So, 24,000 * 1.3478 = 32,347.2 (as calculated earlier)

Then, 24,000 * 0.0000489 = 24,000 * 0.00004 + 24,000 * 0.0000089 = 0.96 + 0.2136 = 1.1736

Therefore, total amount ≈ 32,347.2 + 1.1736 ≈ 32,348.3736.

Therefore, approximately $32,348.37. So, about $32,348.37. Therefore, rounding to the nearest whole number, it would be $32,348.

But wait, let me check if there is any more precise calculation needed. Because depending on the exact value of (1.01)^30, the amount could be slightly more or less. However, given that we have calculated (1.01)^30 ≈ 1.3478489, which is approximately 1.347849, so multiplying by 24,000:

Let me compute 24,000 * 1.347849.

First, note that 24,000 * 1.347849 = 24,000 * 1 + 24,000 * 0.3 + 24,000 * 0.04 + 24,000 * 0.007 + 24,000 * 0.000849.

Wait, breaking down 1.347849 as 1 + 0.3 + 0.04 + 0.007 + 0.000849.

So:

24,000 * 1 = 24,000

24,000 * 0.3 = 7,200

24,000 * 0.04 = 960

24,000 * 0.007 = 168

24,000 * 0.000849 = ?

Compute 24,000 * 0.0008 = 19.2

24,000 * 0.000049 = 24,000 * 0.00004 + 24,000 * 0.000009 = 0.96 + 0.216 = 1.176

Therefore, total for 0.000849 is 19.2 + 1.176 = 20.376

Therefore, adding all together:

24,000 + 7,200 = 31,200

31,200 + 960 = 32,160

32,160 + 168 = 32,328

32,328 + 20.376 = 32,348.376

Therefore, approximately $32,348.38. So, $32,348.38, which rounds to $32,348 when rounded to the nearest whole number.

But wait, let me check if there is a more accurate way. Maybe I can use more decimal places in the exponent?

Alternatively, perhaps I can use the exact value of (1.01)^30. However, without a calculator, it's difficult. However, given that we have calculated it as approximately 1.3478489, which gives us approximately $32,348.38, which is about $32,348.38. Therefore, the nearest whole number is $32,348.

But let me check with another approach. Let me use semi-annual calculations. Wait, but we already did that. Alternatively, maybe I can use the formula for compound interest with more precise decimal places.

Alternatively, perhaps I can use the fact that (1.01)^30 is equal to 1.3478489, so multiplying by 24,000 gives 24,000 * 1.3478489. Let me compute this multiplication more accurately.

Let me write 24,000 * 1.3478489.

First, note that 24,000 * 1.3478489 = 24,000 * 1.3478489.

Let me convert 1.3478489 into a decimal and multiply:

Let me write 1.3478489 as 1.3478489.

Multiplying by 24,000:

First, multiply 1.3478489 by 24,000.

Note that 1.3478489 * 24,000 = 1.3478489 * 24 * 1,000.

Compute 1.3478489 * 24 first.

Compute 1.3478489 * 20 = 26.956978

Compute 1.3478489 * 4 = 5.3913956

Add them together: 26.956978 + 5.3913956 = 32.3483736

Therefore, 1.3478489 * 24 = 32.3483736

Then, multiply by 1,000: 32.3483736 * 1,000 = 32,348.3736

Therefore, the exact value is approximately $32,348.3736, which is approximately $32,348.37 when rounded to the nearest cent. Therefore, when rounded to the nearest whole number, it is $32,348.

But wait, let me check if there is a possibility that the exact value is slightly more than $32,348.50, which would round up to $32,349. However, given that the decimal part is 0.37, which is less than 0.5, it should round down to $32,348.

But let me check once again if my calculation of (1.01)^30 is correct. Because if there is an error in that, then the final answer would be wrong.

Earlier, we calculated (1.01)^10 ≈ 1.1046221, then (1.01)^20 ≈ 1.22019, then (1.01)^30 ≈ 1.22019 * 1.104622 ≈ 1.34785. Let me verify this multiplication again.

Compute 1.22019 * 1.104622:

Let me do this multiplication more carefully.

First, write 1.22019 * 1.104622.

Let me use the standard multiplication method:

Multiply 1.22019 by 1.104622.

First, ignore the decimals and consider them as integers:

But this might be too time-consuming. Alternatively, use the following approach:

Let me approximate 1.22019 * 1.104622.

Let me note that 1.22019 * 1.104622 ≈ 1.22 * 1.1046 ≈ ?

Compute 1.22 * 1.1046:

First, 1 * 1.1046 = 1.1046

0.2 * 1.1046 = 0.22092

0.02 * 1.1046 = 0.022092

Adding them up: 1.1046 + 0.22092 = 1.32552 + 0.022092 = 1.347612

Therefore, approx 1.3476, which is very close to our previous value of 1.34785. Therefore, the exact value is approximately 1.3478, which gives us the total amount as approximately $32,348.37. Therefore, the answer should be $32,348 when rounded to the nearest whole number.

However, to be absolutely sure, let me check with another method. Let me use the formula for compound interest with more precise exponentiation.

Alternatively, I can use the fact that (1.01)^30 can be calculated as follows:

We can use the formula (1 + r)^n = e^{n * ln(1 + r)}.

We already calculated ln(1.01) ≈ 0.00995033, so n * ln(1 + r) = 30 * 0.00995033 ≈ 0.2985099.

Then, e^{0.2985099} ≈ 1.3478489 as calculated before. Therefore, multiplying by 24,000 gives 32,348.37, which rounds to 32,348.

But let me check with another resource or formula. Wait, since I can't actually look up a calculator, but maybe I can use semi-annual calculations step by step for 30 periods. However, that would take a long time, but let me try for a few periods to see if the value is consistent.

For example, let me compute (1.01)^10 ≈ 1.104622 as before.

Then, (1.01)^15 = (1.01)^10 * (1.01)^5 ≈ 1.104622 * 1.051010 ≈ ?

Compute 1.104622 * 1.051010:

First, 1 * 1.051010 = 1.051010

0.1 * 1.051010 = 0.105101

0.004622 * 1.051010 ≈ 0.004855

Adding them up: 1.051010 + 0.105101 = 1.156111 + 0.004855 ≈ 1.160966. Therefore, (1.01)^15 ≈ 1.160966.

Then, (1.01)^20 = (1.01)^15 * (1.01)^5 ≈ 1.160966 * 1.051010 ≈ ?

Compute 1.160966 * 1.051010:

First, 1 * 1.051010 = 1.051010

0.1 * 1.051010 = 0.105101

0.06 * 1.051010 = 0.0630606

0.000966 * 1.051010 ≈ 0.001014

Adding them up:

1.051010 + 0.105101 = 1.156111 + 0.0630606 = 1.2191716 + 0.001014 ≈ 1.2201856. Which matches our previous value of (1.01)^20 ≈ 1.22019. Therefore, that seems correct.

Then, (1.01)^30 = (1.01)^20 * (1.01)^10 ≈ 1.2201856 * 1.104622 ≈ ?

Compute 1.2201856 * 1.104622:

Again, breaking it down:

First, 1 * 1.104622 = 1.104622

0.2 * 1.104622 = 0.2209244

0.02 * 1.104622 = 0.02209244

0.0001856 * 1.104622 ≈ 0.000205

Adding them up:

1.104622 + 0.2209244 = 1.3255464 + 0.02209244 = 1.34763884 + 0.000205 ≈ 1.34784384. Which is approximately 1.347844, which matches our previous value of 1.3478489. Therefore, this confirms that (1.01)^30 ≈ 1.347844.

Therefore, multiplying by 24,000 gives 24,000 * 1.347844 ≈ 32,348.256, which is approximately $32,348.26, which rounds to $32,348.

Therefore, after all these calculations and cross-verifying, I can be confident that the total amount after five years is approximately $32,348 when rounded to the nearest whole number.

However, just to be absolutely thorough, let me check if there's any chance that the interest is compounded differently. The problem says "bi-monthly interest", which means every two months. So, that is, the interest is compounded every two months, which is what I considered. So, the formula A = P*(1 + r)^t is correct here, where r = 0.01, t = 30.

Alternatively, if the problem had said that the annual interest rate is 1% compounded bi-monthly, that would be different. But the problem says "1% bi-monthly interest", meaning that every two months, the investment increases by 1%. Therefore, the rate per period is 1%, and the number of periods is 30. Therefore, the calculation is correct.

Therefore, I think my answer is correct. So, the total amount after five years is approximately $32,348.

But just to make sure, let me check with another approach. Let me use the formula A = P * e^{rt} if it were continuous compounding, but that's not the case here. However, just for comparison, if it were continuous compounding, the amount would be 24,000 * e^{0.01*30} = 24,000 * e^{0.3} ≈ 24,000 * 1.349858 ≈ 32,396.59, which is higher than our value. But since it's compounded discretely every two months, the amount should be less than continuous compounding, which matches our result of ~$32,348, which is less than ~$32,396. So that seems reasonable.

Alternatively, if it were compounded monthly, the amount would be higher than bi-monthly. Wait, no, if it's compounded more frequently, the amount would be higher. Since bi-monthly is every two months, which is less frequent than monthly. Therefore, the amount should be less than if it were compounded monthly. However, since we are compounding every two months, which is less frequent than monthly, so the amount should be less than monthly compounding. However, since we don't have the monthly compounding value, it's hard to compare. But given that our value is ~$32,348, which is less than the continuous compounding value, which is reasonable.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{32348}

To determine the total value of a $24,000 investment after five years with **1% bi-monthly interest**, we need to apply the **compound interest formula**:

$$
A = P(1 + r)^t
$$

Where:
- $ A $ is the final amount,
- $ P = 24,000 $ is the principal,
- $ r = 0.01 $ is the interest rate per period (1%),
- $ t = 30 $ is the number of compounding periods.

---

### Step 1: Determine the Number of Periods

Since the interest is compounded **bi-monthly**, that means it is compounded **every two months**. Over five years:

$$
\text{Number of periods} = \frac{5 \text{ years} \times 12 \text{ months/year}}{2 \text{ months/period}} = 30 \text{ periods}
$$

---

### Step 2: Apply the Compound Interest Formula

$$
A = 24,000 \times (1 + 0.01)^{30} = 24,000 \times (1.01)^{30}
$$

We now compute $ (1.01)^{30} $. Using precise calculation or a calculator:

$$
(1.01)^{30} \approx 1.3478489
$$

---

### Step 3: Multiply by the Principal

$$
A = 24,000 \times 1.3478489 \approx 32,348.37
$$

---

### Step 4: Round to the Nearest Whole Number

$$
\text{Final amount} \approx \boxed{32348}
$$

---

### Final Answer

$$
\boxed{32348}
$$

Solution: Five years make sixty months, so interest will have been compounded 30 times. That means that the investment will have grown to $\$24,\!000 \cdot 1.01^{30} \approx \boxed{\$32,\!348}$, to the nearest dollar.