Unraveling Ellipses: Calculating Horizontal Distance on an Elliptical Arch
Finding Horizontal Distance on an Elliptical Arch
Explore the fascinating world of ellipses as we solve a real-world problem involving an elliptical arch. Imagine an arch 122 feet wide and 33.4 feet high in the shape of an half ellipse. We aim to find the horizontal distance from the center of the arch where the height is equal to 12.4 feet.
Step 1: Understand the Ellipse Parameters
Major Axis Length (2a): 122 feet
Half Minor Axis Length (b): 33.4 feet
Step 2: Formulate the Ellipse Equation
The standard form of the ellipse equation is:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Plugging in the values:
\[
\frac{x^2}{61^2} + \frac{y^2}{33.4^2} = 1
\]
Step 3: Find the Horizontal Distance (\(x\)) for \(y = 12.4\)
Substitute \(y = 12.4\) into the equation and solve for \(x\):
\[
\frac{x^2}{3721} + \frac{12.4^2}{1115.56} = 1
\]
\[
x^2 = \frac{125.13 \times 3721}{1115.56}
\]
\[
x \approx \sqrt{3208.13}
\]
Step 4: Calculate \(x\)
The horizontal distance from the center of the arch where the height is equal to 12.4 feet is approximately 56.6 feet.
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