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The position of the object is given by the function:

\[
s(t) = -t^3 + 3t^2 + 9t + 5
\]

To find the maximum velocity, we first need to find the velocity function by differentiating \( s(t) \) with respect to time \( t \):

\[
v(t) = \frac{ds}{dt} = \frac{d}{dt}(-t^3 + 3t^2 + 9t + 5)
\]

Taking the derivative term by term:

\[
v(t) = -3t^2 + 6t + 9
\]

Now, we need to find the critical points of the velocity function in the interval \([0, 4]\). To do this, we set the derivative of \( v(t) \) equal to zero:

\[
\frac{dv}{dt} = \frac{d}{dt}(-3t^2 + 6t + 9) = -6t + 6
\]

Set this equal to zero and solve for \( t \):

\[
-6t + 6 = 0
\]
\[
t = 1
\]

Now we evaluate the velocity function \( v(t) = -3t^2 + 6t + 9 \) at \( t = 0 \), \( t = 1 \), and \( t = 4 \) (the endpoints of the interval):

\[
v(0) = -3(0)^2 + 6(0) + 9 = 9
\]
\[
v(1) = -3(1)^2 + 6(1) + 9 = 12
\]
\[
v(4) = -3(4)^2 + 6(4) + 9 = -48 + 24 + 9 = -15
\]

The maximum velocity occurs at \( t = 1 \), and the maximum velocity is:

\[
v(1) = 12 \, \text{feet per second}
\]

Thus, the correct answer is:

\[
\boxed{12} \, \text{feet per second}
\]

Would you like more details on any of the steps? 

Here are 5 related questions:
1. How do you determine the critical points of a function?
2. What is the significance of velocity and acceleration in this context?
3. Can you explain why the velocity function involves a derivative of the position function?
4. What is the importance of evaluating velocity at the endpoints of the interval?
5. How would you find the maximum acceleration in the same problem?

**Tip:** Always check both critical points and endpoints when searching for a maximum or minimum in a closed interval!

The position of an object along a vertical line is given by s(t) = -t^3 + 3t^2 + 9t + 5, where s is measured in feet and t is measured in seconds. The maximum velocity of the object in the time interval [0, 4] is:

Learn how to find the maximum velocity of an object using differentiation. This problem involves calculating the derivative of a position function, finding critical points, and evaluating the velocity function at endpoints. Suitable for high school students in Grades 10-12.

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