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The given problem provides the velocity function \( v(t) = -10t + 15t^2 \) and requires us to find \( s(7) \), the position function \( s(t) \), given that \( s(3) = 72 \).

### Steps:
1. **Find the position function \( s(t) \)**:
   The velocity function \( v(t) \) is the derivative of the position function \( s(t) \). Thus, to find \( s(t) \), we need to integrate \( v(t) \).

   \[
   v(t) = \frac{ds}{dt} = -10t + 15t^2
   \]

   To find \( s(t) \), integrate \( v(t) \):

   \[
   s(t) = \int (-10t + 15t^2) dt
   \]

   \[
   s(t) = -5t^2 + 5t^3 + C
   \]

   where \( C \) is the constant of integration.

2. **Use the initial condition** \( s(3) = 72 \) to find \( C \):
   Plug \( t = 3 \) into \( s(t) \) and set it equal to 72:

   \[
   s(3) = -5(3^2) + 5(3^3) + C = 72
   \]

   \[
   -5(9) + 5(27) + C = 72
   \]

   \[
   -45 + 135 + C = 72
   \]

   \[
   90 + C = 72
   \]

   \[
   C = 72 - 90 = -18
   \]

   Thus, the position function is:

   \[
   s(t) = -5t^2 + 5t^3 - 18
   \]

3. **Find \( s(7) \)**:
   Now, substitute \( t = 7 \) into \( s(t) \):

   \[
   s(7) = -5(7^2) + 5(7^3) - 18
   \]

   \[
   s(7) = -5(49) + 5(343) - 18
   \]

   \[
   s(7) = -245 + 1715 - 18
   \]

   \[
   s(7) = 1450 - 18 = 1432
   \]

Thus, \( s(7) = 1432 \).

Would you like any further clarification or details on these steps?

Here are 5 related questions for further practice:
1. How do you find the velocity function if you're given the position function \( s(t) \)?
2. What is the physical meaning of the constant \( C \) when integrating velocity to find position?
3. If \( v(t) \) is a more complex function, does the process of finding \( s(t) \) change?
4. What does it mean if the position function \( s(t) \) is linear?
5. How can you find acceleration from the velocity function?

**Tip**: Always check for initial conditions when integrating, as they are crucial in determining the constants.

The velocity of a particle moving in a straight line is given by v(t) = -10t + 15t^2 where s(3) = 72. Find s(7). (Round your answer to 2 decimal places.)

Solve the problem of finding the position function s(7) when given a velocity function v(t) = -10t + 15t^2, with initial condition s(3) = 72. This solution covers key concepts in calculus such as integration and the application of the Fundamental Theorem of Calculus.

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