A toy rocket is launched vertically into the air from a pad 3 feet above the ground with an initial velocity of 32 feet per second. The height [tex]$ h $[/tex], in feet, of the rocket above the ground at [tex]$ t $[/tex] seconds after launch is given by the function [tex]$ h(t) = -16t^2 + 32t + 3 $[/tex].

1. How long will it take the rocket to reach its maximum height?
2. What is the maximum height?

The rocket reaches its maximum height at [tex]$\square$[/tex] seconds after launch.
(Simplify your answer.)

To solve this problem, we need to determine when the toy rocket reaches its maximum height and what that height is. The height of the rocket at time [tex]$ t $[/tex] seconds is given by the quadratic function:

[tex]\[ h(t) = -16t^2 + 32t + 3 \][/tex]

Step 1: Find the time it takes to reach maximum height.

The formula for the maximum or minimum of a quadratic function [tex]$ ax^2 + bx + c $[/tex] is given by [tex]$ t = -\frac{b}{2a} $[/tex]. In this function:

- [tex]$ a = -16 $[/tex]
- [tex]$ b = 32 $[/tex]

Plug these values into the formula:

[tex]\[ t = -\frac{32}{2 \times -16} = -\frac{32}{-32} = 1 \][/tex]

So, the rocket reaches its maximum height at [tex]$ t = 1 $[/tex] second after launch.

Step 2: Calculate the maximum height.

To find the maximum height, substitute [tex]$ t = 1 $[/tex] back into the height function [tex]$ h(t) = -16t^2 + 32t + 3 $[/tex]:

[tex]\[ h(1) = -16(1)^2 + 32(1) + 3 \][/tex]
[tex]\[ h(1) = -16 \times 1 + 32 \times 1 + 3 \][/tex]
[tex]\[ h(1) = -16 + 32 + 3 \][/tex]
[tex]\[ h(1) = 19 \][/tex]

The maximum height of the rocket is 19 feet.

Therefore, the rocket reaches its maximum height at 1 second after launch, and the maximum height is 19 feet.