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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.
[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.
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