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Answer :
The height of the rocket is given by the quadratic function
[tex]$$
h(t) = -16t^2 + 40t + 4.
$$[/tex]
For a quadratic function of the form
[tex]$$
at^2 + bt + c,
$$[/tex]
the maximum (or minimum) value occurs at
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Here, [tex]$a = -16$[/tex] and [tex]$b = 40$[/tex]. Thus, the time at which the rocket reaches its maximum height is
[tex]$$
t = -\frac{40}{2(-16)} = 1.25 \text{ seconds}.
$$[/tex]
To find the maximum height, we substitute [tex]$t = 1.25$[/tex] back into the height function:
[tex]$$
h(1.25) = -16(1.25)^2 + 40(1.25) + 4.
$$[/tex]
Calculating step by step:
1. Compute [tex]$(1.25)^2$[/tex]:
[tex]$$
(1.25)^2 = 1.5625.
$$[/tex]
2. Multiply by [tex]$-16$[/tex]:
[tex]$$
-16 \times 1.5625 = -25.
$$[/tex]
3. Multiply [tex]$40$[/tex] by [tex]$1.25$[/tex]:
[tex]$$
40 \times 1.25 = 50.
$$[/tex]
4. Sum the results along with [tex]$4$[/tex]:
[tex]$$
-25 + 50 + 4 = 29.
$$[/tex]
Thus, the maximum height reached by the rocket is [tex]$29$[/tex] feet.
The final answers are:
- The rocket reaches its maximum height [tex]$1.25$[/tex] second(s) after launch.
- The maximum height is [tex]$29$[/tex] feet.
[tex]$$
h(t) = -16t^2 + 40t + 4.
$$[/tex]
For a quadratic function of the form
[tex]$$
at^2 + bt + c,
$$[/tex]
the maximum (or minimum) value occurs at
[tex]$$
t = -\frac{b}{2a}.
$$[/tex]
Here, [tex]$a = -16$[/tex] and [tex]$b = 40$[/tex]. Thus, the time at which the rocket reaches its maximum height is
[tex]$$
t = -\frac{40}{2(-16)} = 1.25 \text{ seconds}.
$$[/tex]
To find the maximum height, we substitute [tex]$t = 1.25$[/tex] back into the height function:
[tex]$$
h(1.25) = -16(1.25)^2 + 40(1.25) + 4.
$$[/tex]
Calculating step by step:
1. Compute [tex]$(1.25)^2$[/tex]:
[tex]$$
(1.25)^2 = 1.5625.
$$[/tex]
2. Multiply by [tex]$-16$[/tex]:
[tex]$$
-16 \times 1.5625 = -25.
$$[/tex]
3. Multiply [tex]$40$[/tex] by [tex]$1.25$[/tex]:
[tex]$$
40 \times 1.25 = 50.
$$[/tex]
4. Sum the results along with [tex]$4$[/tex]:
[tex]$$
-25 + 50 + 4 = 29.
$$[/tex]
Thus, the maximum height reached by the rocket is [tex]$29$[/tex] feet.
The final answers are:
- The rocket reaches its maximum height [tex]$1.25$[/tex] second(s) after launch.
- The maximum height is [tex]$29$[/tex] feet.
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