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Answer :
To solve this problem, we need to find the height of the rocket at two specific times: when it is launched (at [tex]\( t = 0 \)[/tex] seconds) and when it returns to the ground (at [tex]\( t = 5 \)[/tex] seconds).
The height of the rocket at any time [tex]\( t \)[/tex] is described by the function:
[tex]\[ f(t) = -16t^2 + 80t \][/tex]
1. Height at launch ([tex]\( t = 0 \)[/tex] seconds):
When the rocket is launched, time [tex]\( t \)[/tex] is 0 seconds. To find the height, we substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
f(0) = -16(0)^2 + 80(0)
\][/tex]
Simplifying this gives:
[tex]\[
f(0) = 0
\][/tex]
So, the height of the rocket at launch is [tex]\( 0 \)[/tex] feet.
2. Height at landing ([tex]\( t = 5 \)[/tex] seconds):
The rocket returns to the ground after [tex]\( t = 5 \)[/tex] seconds. To find the height at this time, we substitute [tex]\( t = 5 \)[/tex] into the function:
[tex]\[
f(5) = -16(5)^2 + 80(5)
\][/tex]
Simplifying this gives:
[tex]\[
f(5) = -16(25) + 400
\][/tex]
[tex]\[
f(5) = -400 + 400
\][/tex]
[tex]\[
f(5) = 0
\][/tex]
So, the height of the rocket when it returns to the ground is [tex]\( 0 \)[/tex] feet.
In conclusion, the rocket is at the ground level (0 feet) both at launch and upon returning after 5 seconds.
The height of the rocket at any time [tex]\( t \)[/tex] is described by the function:
[tex]\[ f(t) = -16t^2 + 80t \][/tex]
1. Height at launch ([tex]\( t = 0 \)[/tex] seconds):
When the rocket is launched, time [tex]\( t \)[/tex] is 0 seconds. To find the height, we substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
f(0) = -16(0)^2 + 80(0)
\][/tex]
Simplifying this gives:
[tex]\[
f(0) = 0
\][/tex]
So, the height of the rocket at launch is [tex]\( 0 \)[/tex] feet.
2. Height at landing ([tex]\( t = 5 \)[/tex] seconds):
The rocket returns to the ground after [tex]\( t = 5 \)[/tex] seconds. To find the height at this time, we substitute [tex]\( t = 5 \)[/tex] into the function:
[tex]\[
f(5) = -16(5)^2 + 80(5)
\][/tex]
Simplifying this gives:
[tex]\[
f(5) = -16(25) + 400
\][/tex]
[tex]\[
f(5) = -400 + 400
\][/tex]
[tex]\[
f(5) = 0
\][/tex]
So, the height of the rocket when it returns to the ground is [tex]\( 0 \)[/tex] feet.
In conclusion, the rocket is at the ground level (0 feet) both at launch and upon returning after 5 seconds.
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