Problem 66 - College Algebra

Question

A bridge is to be built in the shape of a semi- elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

Step-by-Step Solution

Answer

The height at the center is approximately 10.73 feet.

Step 1: Analyze the Problem

We start by understanding that we need to find the height of the semi-elliptical arch at its center. We are given the span (width at the widest part) of the arch and the height at a certain horizontal distance from the center.

Step 2: Represent the Situation Mathematically

The equation of an ellipse centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(2a\) is the span of the arch and \(b\) is the maximum height. In this case, \(2a = 120\) feet, so \(a = 60\).

Step 3: Use Given Information

The height of the arch spans from a point 40 feet horizontally, so \(x = 40\) and the height at this point is 8 feet; thus \(y = 8\). Substitute these values into the ellipse equation: \(\frac{40^2}{60^2} + \frac{8^2}{b^2} = 1\).

Step 4: Simplify and Solve for \(b\)

Calculate \(\frac{40^2}{60^2} = \frac{1600}{3600} = \frac{4}{9}\). Thus, the equation becomes \(\frac{4}{9} + \frac{64}{b^2} = 1\). Solve for \(b^2\) by subtracting \(\frac{4}{9}\) from 1, resulting in \(\frac{5}{9}\), and solve \(\frac{64}{b^2} = \frac{5}{9}\).

Step 5: Calculate \(b\)

Multiply both sides by \(b^2\) and divide by \(\frac{5}{9}\): \(b^2 = \frac{64 \cdot 9}{5}\). This results in \(b^2 = \frac{576}{5}\), so \(b = \sqrt{\frac{576}{5}}\).

Step 6: Conclude the Height at the Center

Compute \(b\): \(b \approx \sqrt{115.2} \approx 10.73\). Therefore, the height of the arch at the center is approximately 10.73 feet.

Key Concepts

Ellipse Equation

The formula for an ellipse is central to solving problems like the semi-elliptical arch. An ellipse is formed by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In this equation, \(a\) and \(b\) are radii along the horizontal and vertical axes respectively. For a semi-ellipse, like the arch of a bridge, we use the upper part of this equation to model the structure.
Understanding this equation allows us to describe the relationship between any point \((x, y)\) on the arch and its corresponding distances from the axes. By substituting known values into this equation, we can find unknown architectural dimensions, such as the height of the arch at its center.

Span and Height Relation

In the context of a semi-elliptical arch, the span represents the total width of the arch, while the height is measured from the center line to the highest point on the arch. The span, denoted as \(2a\), is the total straight-line distance across the arch.
Given the span is 120 feet in our scenario, we determine that \(a = 60\). This establishes the semi-major axis of our ellipse, which is critical in calculating other parameters such as height. The span is directly used in the ellipse equation to define the width of our arch, while the height \(b\) at the center determines the tallest point.

Distance from Center

The given problem includes a specific distance from the center as a data point. Here, we are told the height 8 feet at a distance of 40 feet horizontally from the center.
This piece of information is essential because it provides us the coordinates \((40,8)\) to plug into our ellipse equation. These coordinates help us solve for \(b\), the height at the center of the arch. By knowing part of the structure's geometry, you can establish values for parts you don’t know directly, like the center height of the arch using such geometric principles. It's an application of the ellipse's property of symmetry and its standard relation to distances.

Solving for Variables in Algebraic Expressions

To find the arch's height at its center, we need to solve the equation derived from the ellipse equation for \(b\), the unknown height.
Substituting the given values \((x = 40\), \(y = 8\)) into the ellipse formula \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we derive an expression. The challenge is to isolate \(b\) by performing algebraic manipulations:

• Calculate \(\frac{40^2}{60^2} = \frac{4}{9}\).
• Transform the equation to \(\frac{4}{9} + \frac{64}{b^2} = 1\).
• Solve for \(\frac{64}{b^2}\) to get \(\frac{5}{9}\).
• Cross-multiply to find \(b^2\) and subsequently \(b\).

These steps illustrate how algebra enables us to interpret real-world geometry through precise mathematical calculations. By understanding each manipulation, one can solve for unknown variables efficiently.