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Answer :
We are given the height of the rocket as a function
[tex]$$
h(t) = -16t^2 + 88t + 7.
$$[/tex]
This is a quadratic function, and because the coefficient of [tex]$t^2$[/tex] is negative, the function represents a parabola that opens downward. Therefore, the maximum height is reached at the vertex of the parabola.
The time at which the vertex occurs is given by
[tex]$$
t = -\frac{b}{2a},
$$[/tex]
where [tex]$a = -16$[/tex] and [tex]$b = 88$[/tex]. Substituting these values, we have
[tex]$$
t = -\frac{88}{2(-16)} = \frac{88}{32} = 2.75.
$$[/tex]
So, the rocket reaches its maximum height at [tex]$2.75$[/tex] seconds after launch.
To find the maximum height, substitute [tex]$t = 2.75$[/tex] back into the height equation:
[tex]$$
h(2.75) = -16(2.75)^2 + 88(2.75) + 7.
$$[/tex]
Calculating step by step:
1. Compute [tex]$(2.75)^2$[/tex]:
[tex]$$
(2.75)^2 = 7.5625.
$$[/tex]
2. Multiply by [tex]$-16$[/tex]:
[tex]$$
-16 \times 7.5625 = -121.
$$[/tex]
3. Multiply [tex]$88$[/tex] by [tex]$2.75$[/tex]:
[tex]$$
88 \times 2.75 = 242.
$$[/tex]
4. Now, combine these results with the constant [tex]$7$[/tex]:
[tex]$$
h(2.75) = -121 + 242 + 7 = 128.
$$[/tex]
Thus, the maximum height of the rocket is [tex]$128$[/tex] feet.
In summary, the rocket reaches its maximum height [tex]$2.75$[/tex] seconds after launch and the maximum height is [tex]$128$[/tex] feet.
[tex]$$
h(t) = -16t^2 + 88t + 7.
$$[/tex]
This is a quadratic function, and because the coefficient of [tex]$t^2$[/tex] is negative, the function represents a parabola that opens downward. Therefore, the maximum height is reached at the vertex of the parabola.
The time at which the vertex occurs is given by
[tex]$$
t = -\frac{b}{2a},
$$[/tex]
where [tex]$a = -16$[/tex] and [tex]$b = 88$[/tex]. Substituting these values, we have
[tex]$$
t = -\frac{88}{2(-16)} = \frac{88}{32} = 2.75.
$$[/tex]
So, the rocket reaches its maximum height at [tex]$2.75$[/tex] seconds after launch.
To find the maximum height, substitute [tex]$t = 2.75$[/tex] back into the height equation:
[tex]$$
h(2.75) = -16(2.75)^2 + 88(2.75) + 7.
$$[/tex]
Calculating step by step:
1. Compute [tex]$(2.75)^2$[/tex]:
[tex]$$
(2.75)^2 = 7.5625.
$$[/tex]
2. Multiply by [tex]$-16$[/tex]:
[tex]$$
-16 \times 7.5625 = -121.
$$[/tex]
3. Multiply [tex]$88$[/tex] by [tex]$2.75$[/tex]:
[tex]$$
88 \times 2.75 = 242.
$$[/tex]
4. Now, combine these results with the constant [tex]$7$[/tex]:
[tex]$$
h(2.75) = -121 + 242 + 7 = 128.
$$[/tex]
Thus, the maximum height of the rocket is [tex]$128$[/tex] feet.
In summary, the rocket reaches its maximum height [tex]$2.75$[/tex] seconds after launch and the maximum height is [tex]$128$[/tex] feet.
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