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Answer :
We are given the height of the rocket as a function of time
$$
h(t) = -16t^2 + 128t.
$$
This is a quadratic function in the form
$$
h(t) = at^2 + bt + c,
$$
where in our case \(a = -16\), \(b = 128\), and \(c = 0\).
Since the coefficient \(a\) is negative, the parabola opens downward. This means the vertex of the parabola gives the maximum point of the height.
**Step 1. Find the time at which the maximum height is reached**
The time \(t_{\text{max}}\) at the vertex of a parabola given by
$$
t_{\text{max}} = -\frac{b}{2a}.
$$
Substitute the values for \(a\) and \(b\):
$$
t_{\text{max}} = -\frac{128}{2 \cdot (-16)} = -\frac{128}{-32} = 4.
$$
So, the rocket reaches its maximum height at \(t = 4\) seconds.
**Step 2. Find the maximum height**
Now substitute \(t=4\) back into the height function:
$$
h(4) = -16(4)^2 + 128(4).
$$
First, compute \(4^2\):
$$
4^2 = 16,
$$
so the expression becomes:
$$
h(4) = -16 \cdot 16 + 128 \cdot 4.
$$
Calculate each term:
$$
-16 \cdot 16 = -256,
$$
$$
128 \cdot 4 = 512.
$$
Then, add the two results:
$$
h(4) = -256 + 512 = 256.
$$
Thus, the maximum height reached by the rocket is \(256\) feet.
**Final Answer:**
- The maximum height is \(256\) ft.
- This height is reached at \(4\) seconds.
$$
h(t) = -16t^2 + 128t.
$$
This is a quadratic function in the form
$$
h(t) = at^2 + bt + c,
$$
where in our case \(a = -16\), \(b = 128\), and \(c = 0\).
Since the coefficient \(a\) is negative, the parabola opens downward. This means the vertex of the parabola gives the maximum point of the height.
**Step 1. Find the time at which the maximum height is reached**
The time \(t_{\text{max}}\) at the vertex of a parabola given by
$$
t_{\text{max}} = -\frac{b}{2a}.
$$
Substitute the values for \(a\) and \(b\):
$$
t_{\text{max}} = -\frac{128}{2 \cdot (-16)} = -\frac{128}{-32} = 4.
$$
So, the rocket reaches its maximum height at \(t = 4\) seconds.
**Step 2. Find the maximum height**
Now substitute \(t=4\) back into the height function:
$$
h(4) = -16(4)^2 + 128(4).
$$
First, compute \(4^2\):
$$
4^2 = 16,
$$
so the expression becomes:
$$
h(4) = -16 \cdot 16 + 128 \cdot 4.
$$
Calculate each term:
$$
-16 \cdot 16 = -256,
$$
$$
128 \cdot 4 = 512.
$$
Then, add the two results:
$$
h(4) = -256 + 512 = 256.
$$
Thus, the maximum height reached by the rocket is \(256\) feet.
**Final Answer:**
- The maximum height is \(256\) ft.
- This height is reached at \(4\) seconds.
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