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Part 1
A toy rocket is shot vertically into the air from a launching pad 8 feet above the ground with an initial velocity of 88 feet per second. The height​ h, in​ feet, of the rocket above the ground at t seconds after launch is given by the function h left parenthesis t right parenthesis equals negative 16 t squared plus 88 t plus 8. How long will it take the rocket to reach its maximum​ height? What is the maximum​ height?

The height of the toy rocket at time \( t \) seconds after launch is modeled by the quadratic function:

\[
h(t) = -16t^2 + 88t + 8
\]

### Part 1: How long will it take the rocket to reach its maximum height?

The function \( h(t) \) represents a parabola that opens downward because the coefficient of \( t^2 \) is negative (\( -16 \)). The time \( t \) at which the rocket reaches its maximum height is given by the vertex of the parabola. The formula to find the time at which the vertex occurs for a quadratic equation \( ax^2 + bx + c \) is:

\[
t_{\text{max}} = \frac{-b}{2a}
\]

For the given function \( h(t) = -16t^2 + 88t + 8 \), the coefficients are:
- \( a = -16 \)
- \( b = 88 \)

Using the formula:

\[
t_{\text{max}} = \frac{-88}{2(-16)} = \frac{-88}{-32} = 2.75 \text{ seconds}
\]

So, it will take **2.75 seconds** for the rocket to reach its maximum height.

### Part 2: What is the maximum height?

To find the maximum height, substitute \( t = 2.75 \) back into the height function \( h(t) \):

\[
h(2.75) = -16(2.75)^2 + 88(2.75) + 8
\]

First, calculate \( (2.75)^2 \):

\[
(2.75)^2 = 7.5625
\]

Now, substitute into the equation:

\[
h(2.75) = -16(7.5625) + 88(2.75) + 8
\]

\[
h(2.75) = -121 + 242 + 8
\]

\[
h(2.75) = 129 \text{ feet}
\]

Thus, the maximum height of the rocket is **129 feet**.

### Summary:
- Time to reach maximum height: **2.75 seconds**
- Maximum height: **129 feet**

Would you like more details or have any other questions?

Here are 5 related questions:
1. What would happen to the maximum height if the initial velocity were increased?
2. How would the time to reach the maximum height change with a different initial velocity?
3. Can you find the time when the rocket hits the ground?
4. What is the significance of the initial height in this equation?
5. What does the negative coefficient of \( t^2 \) tell us about the motion?

**Tip:** To find the vertex of any quadratic function, use \( t_{\text{max}} = \frac{-b}{2a} \), where the quadratic is in the form \( ax^2 + bx + c \).

A toy rocket is shot vertically into the air from a launching pad 8 feet above the ground with an initial velocity of 88 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t) = -16t^2 + 88t + 8. How long will it take the rocket to reach its maximum height? What is the maximum height?

Solve the problem of finding how long it takes a toy rocket to reach its maximum height and what that height is. Learn to use the vertex formula for quadratic functions to calculate both the time and height. This problem involves algebra and kinematic principles, suitable for Grades 9-11.

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