Step 1: Understanding the Question: We need to model the elliptical arch with a mathematical equation.
The horizontal span gives us the major axis, and the maximum height gives us the semi-minor axis.
We then need to find the height of the arch at a specific horizontal distance from the center.
Finally, we calculate the difference between the height of the road and the height of the arch at that point.
Step 2: Key Formula or Approach:
The standard equation of an ellipse centered at the origin (0,0) is:
[ frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ]
Here, (2a) is the length of the major axis (horizontal span) and (b) is the length of the semi-minor axis (maximum height).
Step 3: Detailed Explanation:
1. Determine the parameters of the ellipse:
The horizontal span is 50 ft, which corresponds to the major axis (2a).
[ 2a = 50 implies a = 25 , text{ft} ]
The maximum height of the arch is 20 ft, which corresponds to the semi-minor axis (b).
[ b = 20 , text{ft} ]
So, the equation of the elliptical arch is:
[ frac{x^2}{25^2} + frac{y^2}{20^2} = 1 ]
2. Find the height of the arch at the given point:
We need to find the height (y) of the arch at a point that is 15 ft away from the center horizontally. This means we set (x = 15).
[ frac{15^2}{25^2} + frac{y^2}{20^2} = 1 ]
[ frac{225}{625} + frac{y^2}{400} = 1 ]
Simplifying the fraction: [ frac{9 times 25}{25 times 25} = frac{9}{25} ]
So the equation becomes:
[ frac{9}{25} + frac{y^2}{400} = 1 ]
Now, solve for (y^2): [ frac{y^2}{400} = 1 - frac{9}{25} = frac{25 - 9}{25} = frac{16}{25} ]
[ y^2 = 400 times frac{16}{25} = (16 times 25) times frac{16}{25} = 16 times 16 = 256 ]
[ y = sqrt{256} = 16 , text{ft} ]
This is the vertical height of the arch 15 ft from the center.
3. Determine the height of the bridge road:
The highest point of the arch is its maximum height, which is (b = 20) ft.
The bridge road is constructed 4 ft above this highest point.
[ text{Road Height} = 20 , text{ft} + 4 , text{ft} = 24 , text{ft} ]
4. Calculate the required vertical distance:
The vertical distance of the point on the arch from the bridge road is the difference between the road's height and the arch's height at (x = 15).
[ text{Distance} = text{Road Height} - y = 24 , text{ft} - 16 , text{ft} = 8 , text{ft} ]
Step 4: Final Answer:
The vertical distance of the point from the bridge road is 8 ft.