To find the position function $s(t)$ from the given velocity function $v(t) = 7 \cos(t) + 2 \sin(t)$ and initial condition $s(0) = 4$, we need to integrate the velocity function with respect to $t$.

Solution:

1. Integrate $v(t)$ to find $s(t)$:

   $s(t) = \int v(t) \, dt = \int (7 \cos(t) + 2 \sin(t)) \, dt$

2. Compute the integral:

   • The integral of $7 \cos(t)$ is $7 \sin(t)$.
   • The integral of $2 \sin(t)$ is $-2 \cos(t)$.

   So, we get:

   $s(t) = 7 \sin(t) - 2 \cos(t) + C$

3. Determine the constant $C$ using the initial condition $s(0) = 4$:

   Substitute $t = 0$ and $s(0) = 4$:

   $s(0) = 7 \sin(0) - 2 \cos(0) + C = 4$

   Since $\sin(0) = 0$ and $\cos(0) = 1$, we have:

   $0 - 2 + C = 4$

   $C = 6$

4. Write the final position function:

   $s(t) = 7 \sin(t) - 2 \cos(t) + 6$

Answer:

$s(t) = 7 \sin(t) - 2 \cos(t) + 6$

Would you like a more detailed explanation of any step?

Here are some related questions that might help expand your understanding:

1. How would you find the position function if the initial condition was different, say $s(0) = 10$?
2. What is the physical meaning of the constant $C$ in this context?
3. How does the integration of velocity relate to displacement and position in motion problems?
4. How would the solution change if $v(t)$ included a $t$-dependent term like $t \cos(t)$?
5. What are some other real-world applications of integrating a velocity function?

Tip: Always remember that integration of a velocity function yields the position function, with an arbitrary constant that’s determined by initial conditions.