This problem is asking to find the position function $s(t)$ of an object moving along a line, given the following information:

1. The acceleration is $a(t) = \frac{d^2 s}{dt^2} = \frac{64}{\pi^2} \cos\left(\frac{8t}{\pi}\right)$.
2. The initial velocity is $v(0) = 4$ and the initial position is $s(0) = 5$.

To solve for the position function $s(t)$, we need to follow these steps:

Step 1: Find the velocity function $v(t)$

We integrate the acceleration $a(t)$ to find the velocity function.

$v(t) = \int a(t) \, dt = \int \frac{64}{\pi^2} \cos\left(\frac{8t}{\pi}\right) \, dt$

We can use the fact that the integral of $\cos(kx)$ is $\frac{1}{k} \sin(kx)$, where $k$ is the coefficient of $x$. In this case, $k = \frac{8}{\pi}$.

So,

$v(t) = \frac{64}{\pi^2} \cdot \frac{\pi}{8} \sin\left(\frac{8t}{\pi}\right) + C_1$ $v(t) = \frac{8}{\pi} \sin\left(\frac{8t}{\pi}\right) + C_1$

Step 2: Use initial condition $v(0) = 4$

To find the constant $C_1$, use the initial condition $v(0) = 4$.

At $t = 0$: $v(0) = \frac{8}{\pi} \sin\left(\frac{8(0)}{\pi}\right) + C_1 = 4$ $\sin(0) = 0, \quad \text{so} \quad C_1 = 4$

Thus, the velocity function becomes: $v(t) = \frac{8}{\pi} \sin\left(\frac{8t}{\pi}\right) + 4$

Step 3: Find the position function $s(t)$

Now, integrate $v(t)$ to find the position function $s(t)$:

$s(t) = \int v(t) \, dt = \int \left( \frac{8}{\pi} \sin\left(\frac{8t}{\pi}\right) + 4 \right) dt$

The integral of $\sin(kx)$ is $-\frac{1}{k} \cos(kx)$, so:

$s(t) = -\frac{8}{\pi} \cdot \frac{\pi}{8} \cos\left(\frac{8t}{\pi}\right) + 4t + C_2$ $s(t) = -\cos\left(\frac{8t}{\pi}\right) + 4t + C_2$

Step 4: Use initial condition $s(0) = 5$

To find the constant $C_2$, use the initial condition $s(0) = 5$.

At $t = 0$: $s(0) = -\cos(0) + 4(0) + C_2 = 5$ $-\cos(0) = -1, \quad \text{so} \quad -1 + C_2 = 5$ $C_2 = 6$

Thus, the position function is: $s(t) = -\cos\left(\frac{8t}{\pi}\right) + 4t + 6$

Final Answer:

$s(t) = -\cos\left(\frac{8t}{\pi}\right) + 4t + 6$

Would you like more details on this solution or have any questions?

Here are some related questions to extend your understanding:

1. How would you find the velocity at any given time $t$?
2. What is the object's acceleration at $t = 0$?
3. How would the problem change if the initial velocity was zero?
4. Can you describe the motion of the object based on the cosine term in the position function?
5. How does the amplitude of the cosine function affect the object's motion?

Tip: Always check your integration constants by using the given initial conditions to ensure accuracy.