SOLUTION: arch in an amusement park has a parabolic shape. it has a height of 74 feet and the base width of 38 feet. what is the height of the Arch 8 feet from the center?

Question 1165709: arch in an amusement park has a parabolic shape. it has a height of 74 feet and the base width of 38 feet. what is the height of the Arch 8 feet from the center?
Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=2
# Given parameters
H = 74 # Maximum height (feet)
W = 38 # Base width (feet)
x = 8 # Horizontal distance from the center (feet)
# The parabola is defined by y = a*x^2 + H.
# The base endpoints are at (W/2, 0) and (-W/2, 0).
# W/2 = 38 / 2 = 19
# We use the point (19, 0) to find 'a'.
# 0 = a * (19)^2 + 74
# a = -74 / 19^2 = -74 / 361
# The equation of the parabola is: y = (-74/361) * x^2 + 74
# Calculate the height (y) at x = 8
height = (-74 / 361) * (x**2) + H
print(f"Height at 8 feet from the center: {height} feet")
```
```text?code_stdout&code_event_index=2
Height at 8 feet from the center: 60.880886426592795 feet
```
The height of the arch 8 feet from the center is approximately **60.88 feet**.
Here is the step-by-step calculation:
### 1\. Establish the Equation of the Parabola
We model the parabolic arch using a standard coordinate system where the vertex (maximum height) is on the $y$-axis and the base is on the $x$-axis.
The general equation for a parabola opening downward with a vertex at $(0, h)$ is:
$$y = ax^2 + h$$
* **Height ($h$):** $74$ feet.
* **Base Width:** $38$ feet, meaning the arch touches the ground (where $y=0$) at $x = \pm \frac{38}{2} = \pm 19$.
Using the point $(19, 0)$ and the vertex $h=74$:
$$0 = a(19)^2 + 74$$
$$0 = 361a + 74$$
$$a = -\frac{74}{361}$$
The equation for the arch is:
$$y = -\frac{74}{361}x^2 + 74$$
### 2\. Calculate the Height
To find the height $y$ at a horizontal distance of $x=8$ feet from the center:
$$y = -\frac{74}{361}(8)^2 + 74$$
$$y = -\frac{74}{361}(64) + 74$$
$$y = 74 - \frac{4736}{361}$$
$$y \approx 74 - 13.11911$$
$$y \approx 60.88088 \text{ feet}$$
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