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Answer :
We are given the function
[tex]$$
f(t) = -16t^2 + 80t,
$$[/tex]
which models the height (in feet) of the rocket at time [tex]$t$[/tex] (in seconds). To find the maximum height of the rocket, we need to determine the vertex of this quadratic function.
A quadratic function in the form
[tex]$$
f(t) = at^2 + bt + c
$$[/tex]
reaches its maximum (or minimum) at
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Here, the coefficients are:
[tex]$$
a = -16 \quad \text{and} \quad b = 80.
$$[/tex]
Substitute these into the formula to find the time at which the maximum height occurs:
[tex]$$
t = -\frac{80}{2(-16)} = -\frac{80}{-32} = 2.5.
$$[/tex]
Now, substitute [tex]$t = 2.5$[/tex] back into the original function to find the maximum height:
[tex]$$
f(2.5) = -16(2.5)^2 + 80(2.5).
$$[/tex]
First, calculate [tex]$(2.5)^2$[/tex]:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Then, substitute:
[tex]$$
f(2.5) = -16(6.25) + 80(2.5) = -100 + 200 = 100.
$$[/tex]
Thus, the maximum height of the rocket is
[tex]$$
\boxed{100 \text{ feet}}.
$$[/tex]
[tex]$$
f(t) = -16t^2 + 80t,
$$[/tex]
which models the height (in feet) of the rocket at time [tex]$t$[/tex] (in seconds). To find the maximum height of the rocket, we need to determine the vertex of this quadratic function.
A quadratic function in the form
[tex]$$
f(t) = at^2 + bt + c
$$[/tex]
reaches its maximum (or minimum) at
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Here, the coefficients are:
[tex]$$
a = -16 \quad \text{and} \quad b = 80.
$$[/tex]
Substitute these into the formula to find the time at which the maximum height occurs:
[tex]$$
t = -\frac{80}{2(-16)} = -\frac{80}{-32} = 2.5.
$$[/tex]
Now, substitute [tex]$t = 2.5$[/tex] back into the original function to find the maximum height:
[tex]$$
f(2.5) = -16(2.5)^2 + 80(2.5).
$$[/tex]
First, calculate [tex]$(2.5)^2$[/tex]:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Then, substitute:
[tex]$$
f(2.5) = -16(6.25) + 80(2.5) = -100 + 200 = 100.
$$[/tex]
Thus, the maximum height of the rocket is
[tex]$$
\boxed{100 \text{ feet}}.
$$[/tex]
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