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Answer :
To solve this problem and find out when the toy rocket reaches its maximum height and what that maximum height is, we need to look at the function that describes the height of the rocket over time:
[tex]\[ h(t) = -16t^2 + 64t + 5 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 64 \)[/tex], and [tex]\( c = 5 \)[/tex].
### Find the Time to Reach Maximum Height
The maximum height of a parabolic function occurs at its vertex. The formula to find the time at which a parabola reaches its vertex is given by:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ t = \frac{-64}{2 \times -16} \][/tex]
[tex]\[ t = \frac{-64}{-32} \][/tex]
[tex]\[ t = 2 \][/tex]
So, the rocket reaches its maximum height 2 seconds after it is launched.
### Calculate the Maximum Height
To find the maximum height, substitute [tex]\( t = 2 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(2) = -16(2)^2 + 64(2) + 5 \][/tex]
First, calculate [tex]\( (2)^2 \)[/tex]:
[tex]\[ (2)^2 = 4 \][/tex]
Then multiply by [tex]\(-16\)[/tex]:
[tex]\[ -16 \times 4 = -64 \][/tex]
Now calculate [tex]\( 64 \times 2 \)[/tex]:
[tex]\[ 64 \times 2 = 128 \][/tex]
Add these results together with the constant term 5:
[tex]\[ -64 + 128 + 5 = 69 \][/tex]
Thus, the maximum height the rocket reaches is 69 feet.
In summary, the rocket reaches its maximum height of 69 feet at 2 seconds after launch.
[tex]\[ h(t) = -16t^2 + 64t + 5 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 64 \)[/tex], and [tex]\( c = 5 \)[/tex].
### Find the Time to Reach Maximum Height
The maximum height of a parabolic function occurs at its vertex. The formula to find the time at which a parabola reaches its vertex is given by:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ t = \frac{-64}{2 \times -16} \][/tex]
[tex]\[ t = \frac{-64}{-32} \][/tex]
[tex]\[ t = 2 \][/tex]
So, the rocket reaches its maximum height 2 seconds after it is launched.
### Calculate the Maximum Height
To find the maximum height, substitute [tex]\( t = 2 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[ h(2) = -16(2)^2 + 64(2) + 5 \][/tex]
First, calculate [tex]\( (2)^2 \)[/tex]:
[tex]\[ (2)^2 = 4 \][/tex]
Then multiply by [tex]\(-16\)[/tex]:
[tex]\[ -16 \times 4 = -64 \][/tex]
Now calculate [tex]\( 64 \times 2 \)[/tex]:
[tex]\[ 64 \times 2 = 128 \][/tex]
Add these results together with the constant term 5:
[tex]\[ -64 + 128 + 5 = 69 \][/tex]
Thus, the maximum height the rocket reaches is 69 feet.
In summary, the rocket reaches its maximum height of 69 feet at 2 seconds after launch.
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