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Answer :
The height of the rocket is given by the quadratic function
[tex]$$
f(t) = -16t^2 + 80t.
$$[/tex]
Because the coefficient of [tex]$t^2$[/tex] is negative, the graph of [tex]$f(t)$[/tex] is a downward‐opening parabola, which means its highest point is at the vertex.
To find the time [tex]$t_{\text{max}}$[/tex] when the rocket reaches its maximum height, we use the formula for the vertex of a quadratic function in the form [tex]$at^2 + bt + c$[/tex]:
[tex]$$
t_{\text{max}} = -\frac{b}{2a}.
$$[/tex]
Here, [tex]$a = -16$[/tex] and [tex]$b = 80$[/tex]. Substituting these values in, we get
[tex]$$
t_{\text{max}} = -\frac{80}{2(-16)} = -\frac{80}{-32} = 2.5 \text{ seconds}.
$$[/tex]
Next, substitute [tex]$t_{\text{max}} = 2.5$[/tex] back into the height function to find the maximum height [tex]$h_{\text{max}}$[/tex]:
[tex]$$
h_{\text{max}} = -16(2.5)^2 + 80(2.5).
$$[/tex]
Compute [tex]$(2.5)^2$[/tex]:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Now, calculate:
[tex]$$
h_{\text{max}} = -16(6.25) + 80(2.5) = -100 + 200 = 100 \text{ feet}.
$$[/tex]
Thus, the maximum height of the rocket is [tex]$\boxed{100}$[/tex] feet, and it reaches this height at [tex]$\boxed{2.5}$[/tex] seconds.
[tex]$$
f(t) = -16t^2 + 80t.
$$[/tex]
Because the coefficient of [tex]$t^2$[/tex] is negative, the graph of [tex]$f(t)$[/tex] is a downward‐opening parabola, which means its highest point is at the vertex.
To find the time [tex]$t_{\text{max}}$[/tex] when the rocket reaches its maximum height, we use the formula for the vertex of a quadratic function in the form [tex]$at^2 + bt + c$[/tex]:
[tex]$$
t_{\text{max}} = -\frac{b}{2a}.
$$[/tex]
Here, [tex]$a = -16$[/tex] and [tex]$b = 80$[/tex]. Substituting these values in, we get
[tex]$$
t_{\text{max}} = -\frac{80}{2(-16)} = -\frac{80}{-32} = 2.5 \text{ seconds}.
$$[/tex]
Next, substitute [tex]$t_{\text{max}} = 2.5$[/tex] back into the height function to find the maximum height [tex]$h_{\text{max}}$[/tex]:
[tex]$$
h_{\text{max}} = -16(2.5)^2 + 80(2.5).
$$[/tex]
Compute [tex]$(2.5)^2$[/tex]:
[tex]$$
(2.5)^2 = 6.25.
$$[/tex]
Now, calculate:
[tex]$$
h_{\text{max}} = -16(6.25) + 80(2.5) = -100 + 200 = 100 \text{ feet}.
$$[/tex]
Thus, the maximum height of the rocket is [tex]$\boxed{100}$[/tex] feet, and it reaches this height at [tex]$\boxed{2.5}$[/tex] seconds.
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