Question 1210513: Scientists launched a rocket which can be modeled by the equation h = -16t2 + 160t + 1, where t is time in seconds, and h is the distance from the ground in feet.
a) Find the maximum height of this rocket.
____ feet
b) When does the maximum height occur?
_ seconds
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! This problem can be solved by finding the vertex of the parabola described by the height function:
$$h(t) = -16t^2 + 260t + 1$$
This is a quadratic equation in the form $h(t) = at^2 + bt + c$, where $a = -16$, $b = 260$, and $c = 1$. Since $a$ is negative, the parabola opens downward, and the vertex represents the maximum height.
### a) Find the maximum height of this rocket.
The maximum height occurs at the time $t$ given by the axis of symmetry formula:
$$t = -\frac{b}{2a}$$
The maximum height ($h_{\text{max}}$) is found by substituting this time back into the height equation $h(t)$.
#### Step 1: Find the time of maximum height ($t$)
$$t = -\frac{260}{2(-16)} = -\frac{260}{-32} = \frac{260}{32}$$
$$t = 8.125 \text{ seconds}$$
#### Step 2: Calculate the maximum height ($h$)
Substitute $t = 8.125$ into the height equation:
$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$
I will use the code interpreter to perform the calculation precisely.
```python?code_reference&code_event_index=2
# Given function parameters
a = -16
b = 260
c = 1
# Calculate time (t) of maximum height (t = -b / 2a)
t_max = -b / (2 * a)
# Calculate the maximum height (h) by substituting t_max into the equation h = at^2 + bt + c
h_max = a * (t_max**2) + b * t_max + c
print(f"Time of maximum height (t_max): {t_max}")
print(f"Maximum height (h_max): {h_max}")
```
```text?code_stdout&code_event_index=2
Time of maximum height (t_max): 8.125
Maximum height (h_max): 1057.25
```
### a) Find the maximum height of this rocket.
Substituting $t = 8.125$ into the equation:
$$h_{\text{max}} = -16(8.125)^2 + 260(8.125) + 1$$
$$\mathbf{h_{\text{max}} = 1057.25 \text{ feet}}$$
### b) When does the maximum height occur?
The maximum height occurs at the time calculated by the axis of symmetry formula:
$$t = -\frac{b}{2a} = 8.125$$
$$\mathbf{t = 8.125 \text{ seconds}}$$
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**Final Answers:**
a) Find the maximum height of this rocket.
**1057.25** feet
b) When does the maximum height occur?
**8.125** seconds
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