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Answer :
To determine how long it will take for the rocket to reach its maximum height and what that maximum height will be, we can analyze the function representing the height of the rocket over time: [tex]\( h(t) = -16t^2 + 152t + 5 \)[/tex].
1. Determine the time to reach maximum height:
The height function [tex]\( h(t) = -16t^2 + 152t + 5 \)[/tex] is a quadratic equation in the form [tex]\( At^2 + Bt + C \)[/tex], where [tex]\( A = -16 \)[/tex], [tex]\( B = 152 \)[/tex], and [tex]\( C = 5 \)[/tex].
To find the time at which the maximum height occurs, we use the vertex formula for a parabola. The vertex of a parabola given by [tex]\( At^2 + Bt + C \)[/tex] occurs at:
[tex]\[
t = -\frac{B}{2A}
\][/tex]
Plugging in the values for [tex]\( B \)[/tex] and [tex]\( A \)[/tex]:
[tex]\[
t = -\frac{152}{2(-16)} = \frac{152}{32} = 4.75 \text{ seconds}
\][/tex]
Therefore, the rocket reaches its maximum height [tex]\( 4.75 \)[/tex] seconds after launch.
2. Determine the maximum height:
To find the maximum height, we substitute [tex]\( t = 4.75 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[
h(4.75) = -16(4.75)^2 + 152(4.75) + 5
\][/tex]
After calculating the above expression, the result is:
[tex]\[
h(4.75) = 366 \text{ feet}
\][/tex]
Thus, the maximum height the rocket reaches is [tex]\( 366 \)[/tex] feet.
In summary:
- The rocket reaches its maximum height at [tex]\( 4.75 \)[/tex] seconds after launch.
- The maximum height is [tex]\( 366 \)[/tex] feet.
1. Determine the time to reach maximum height:
The height function [tex]\( h(t) = -16t^2 + 152t + 5 \)[/tex] is a quadratic equation in the form [tex]\( At^2 + Bt + C \)[/tex], where [tex]\( A = -16 \)[/tex], [tex]\( B = 152 \)[/tex], and [tex]\( C = 5 \)[/tex].
To find the time at which the maximum height occurs, we use the vertex formula for a parabola. The vertex of a parabola given by [tex]\( At^2 + Bt + C \)[/tex] occurs at:
[tex]\[
t = -\frac{B}{2A}
\][/tex]
Plugging in the values for [tex]\( B \)[/tex] and [tex]\( A \)[/tex]:
[tex]\[
t = -\frac{152}{2(-16)} = \frac{152}{32} = 4.75 \text{ seconds}
\][/tex]
Therefore, the rocket reaches its maximum height [tex]\( 4.75 \)[/tex] seconds after launch.
2. Determine the maximum height:
To find the maximum height, we substitute [tex]\( t = 4.75 \)[/tex] back into the height function [tex]\( h(t) \)[/tex]:
[tex]\[
h(4.75) = -16(4.75)^2 + 152(4.75) + 5
\][/tex]
After calculating the above expression, the result is:
[tex]\[
h(4.75) = 366 \text{ feet}
\][/tex]
Thus, the maximum height the rocket reaches is [tex]\( 366 \)[/tex] feet.
In summary:
- The rocket reaches its maximum height at [tex]\( 4.75 \)[/tex] seconds after launch.
- The maximum height is [tex]\( 366 \)[/tex] feet.
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