To find the position function $s(t)$, we need to integrate the velocity function $v(t) = 7t^4$, and then use the given condition $s(0) = 8$ to determine the constant of integration.

Step 1: Integrate the velocity function

$v(t) = 7t^4$ The indefinite integral is: $\int 7t^4 \, dt = \frac{7}{5}t^5 + C$ where $C$ is the constant of integration.

Step 2: Solve for $C$ using $s(0) = 8$

The position function is: $s(t) = \frac{7}{5}t^5 + C$ Substitute $t = 0$ and $s(0) = 8$: $s(0) = \frac{7}{5}(0)^5 + C = 8$ $C = 8$

Step 3: Write the position function

$s(t) = \frac{7}{5}t^5 + 8$

Final Answer:

The correct choice is: $\text{D. } s(t) = \frac{7}{5}t^5 + 8$

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Would you like further clarification or a breakdown of any specific step?

Here are 5 related questions to explore further:

1. What is the physical meaning of the constant of integration in position functions?
2. How would you find acceleration from the given velocity function?
3. If the velocity function were $v(t) = 7t^3$, how would the position function change?
4. How does the initial condition $s(0)$ affect the solution?
5. What is the difference between definite and indefinite integration?

Tip: Always remember to apply initial conditions after integrating to solve for the constant of integration!