Problem 65 - College Algebra

Question

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

Step-by-Step Solution

Answer

The distance from the center is approximately 17.32 feet.

Step 1: Understand the Problem

The problem describes a semi-ellipse arch with a height of 12 feet and a span of 40 feet. We need to find the equation of the ellipse and determine the distance from the center to a point where the height is 6 feet.

Step 2: Find the Ellipse Equation

For a semi-ellipse, use the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) is half the span and \( b \) is the height. Here, \( a = 20 \) and \( b = 12 \). Thus, the equation of the ellipse becomes: \[ \frac{x^2}{20^2} + \frac{y^2}{12^2} = 1 \], simplifying to \[ \frac{x^2}{400} + \frac{y^2}{144} = 1 \].

Step 3: Set the Height to 6 Feet

Substitute \( y = 6 \) into the ellipse equation \( \frac{x^2}{400} + \frac{y^2}{144} = 1 \) to find \( x \). Thus, we have: \[ \frac{x^2}{400} + \frac{6^2}{144} = 1 \].

Step 4: Solve for x

Equation simplifies to \( \frac{x^2}{400} + \frac{36}{144} = 1 \). Since \( \frac{36}{144} = 0.25 \), rewrite as \( \frac{x^2}{400} = 0.75 \). Multiply both sides by 400 to solve for \( x^2 \): \( x^2 = 300 \).

Step 5: Calculate x and Provide the Distance

Take the square root of both sides to find \( x \): \( x = \sqrt{300} \). Calculate \( \sqrt{300} \approx 17.32 \). This is the distance from the center to the point where the height is 6 feet.

Key Concepts

Understanding the Semi-Ellipse

A semi-ellipse is half of an ellipse, usually represented above a horizontal line. The semi-ellipse shares the same major and minor axes halves as a complete ellipse. In this case, the problem involves an arch shaped like a semi-ellipse, which means only the upper portion of the ellipse is considered.
The term "span" refers to the total horizontal distance across the base of the semi-ellipse. Here, the span is 40 feet. The "height" refers to the maximum vertical distance from the base to the highest point.

• For a full ellipse, the length of the major axis is the total span, but for a semi-ellipse, it is halved.
• The height represents the semi-minor axis length in our equation.

Understanding these terms helps in forming the equation for the semi-ellipse, which is crucial for solving the problem.

Basics of Coordinate Geometry in Ellipse

Coordinate geometry is essential when dealing with ellipses, as positions and lengths are defined using coordinates. An ellipse central alignment is set on a Cartesian coordinate system, which simplifies calculations and derivation of equations. The formula to describe an ellipse is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\]where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. In this scenario:

• \(a\) equals half of the total span (20 feet)
• \(b\) equals the arch's height (12 feet)

Now, the ellipse's equation becomes: \[ \frac{x^2}{400} + \frac{y^2}{144} = 1.\]This foundational understanding of coordinate geometry allows us to locate specific points on the semi-ellipse and solve for specific coordinates according to given conditions.

Steps of Distance Calculation

Distance calculation on an ellipse often requires substituting known values into its equation to solve for unknowns. In our context, we are trying to find the horizontal distance from the center point of the semi-ellipse to a point where the vertical height (\(y\)) is 6 feet. Using the calculated ellipse equation:
Substituting \(y = 6\) into the equation gives us:
\[ \frac{x^2}{400} + \frac{6^2}{144} = 1,\]which simplifies to:\[ \frac{x^2}{400} + 0.25 = 1\]This further simplifies to:\[ \frac{x^2}{400} = 0.75.\]Multiplying by 400 to solve for \(x^2\):\[ x^2 = 300\]By taking the square root, we find:\[ x = \sqrt{300} \approx 17.32\]Thus, the horizontal distance from the center to the specified point on the semi-ellipse is approximately 17.32 feet. This approach showcases how understanding the basic principles of coordinate geometry leads to effective distance calculations.