Problem 55 - Algebra and Trigonometry

Question

Sunburst Window A “sunburst” window above a door-way is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its highest point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.

Step-by-Step Solution

Answer

The height of the window 25 in. from the center is approximately 15.61 in.

Step 1: Identifying the Major and Minor Axes

The given ellipse's top half represents the sunburst window. In an ellipse, the maximum width is given by its major axis, and the height by its minor axis. Here, the total width (major axis) is 80 in, so the semi-major axis is \( a = \frac{80}{2} = 40 \) in. The maximum height (minor axis) is 20 in, hence the semi-minor axis is \( b = 20 \) in.

Step 2: Equation of the Ellipse

The standard equation of an ellipse centered at the origin with semi-major axis \( a \) and semi-minor axis \( b \) is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Substituting the given values, the equation becomes \( \frac{x^2}{40^2} + \frac{y^2}{20^2} = 1 \) or \( \frac{x^2}{1600} + \frac{y^2}{400} = 1 \).

Step 3: Substituting the x-value

We need to find the height \( y \) at 25 in. from the center. Substitute \( x = 25 \) into the ellipse equation: \( \frac{25^2}{1600} + \frac{y^2}{400} = 1 \).

Step 4: Solving for y

Calculate \( \frac{25^2}{1600} = \frac{625}{1600} = 0.390625 \). Substitute this value into the equation: \( 0.390625 + \frac{y^2}{400} = 1 \). Rearrange to solve for \( y^2 \): \( \frac{y^2}{400} = 1 - 0.390625 = 0.609375 \). Then, \( y^2 = 400 \times 0.609375 = 243.75 \). Taking the square root, \( y = \sqrt{243.75} \approx 15.61 \) in.

Key Concepts

Semi-Major Axis

The semi-major axis is one of the most important characteristics of an ellipse. It represents half the length of the longest diameter of the ellipse. You can think of it as the "radius" of the ellipse in the direction of the major (longest) axis.
In the context of the sunburst window exercise, the total width of the window is 80 inches. Since the major axis spans the widest part of the ellipse, we find the semi-major axis by halving this width. Thus, the semi-major axis measures 40 inches.
This semi-major axis helps us to scale and define the ellipse in terms of its widest reach from the center point, helping in further calculations like determining the height at a specific width.

Semi-Minor Axis

The semi-minor axis, on the other hand, describes half of the shortest diameter of the ellipse. It is perpendicular to the semi-major axis and serves as the "radius" in the shortest direction across the ellipse.
For the sunburst window, the maximum height we can measure from the base to the top is 20 inches. Therefore, the length of the semi-minor axis is 20 inches. This axis defines the curvature of the ellipse between the endpoints of the semi-major axis.
Understanding the semi-minor axis is crucial in calculating the form of the ellipse and understanding its different aspects, such as how it influences the shape's overall height and curvature.

Ellipse Properties

Ellipses have unique properties that distinguish them from other shapes. Two key properties are their reflections and axis symmetry.
1. **Reflection Properties**: All ellipses reflect light in an interesting way. A ray of light emanating from one focus will reflect back to the other focus. This property is used in designing things like satellite dishes.
2. **Symmetry**: An ellipse is symmetric about its major and minor axes. This means that if you were to fold an ellipse along either axis, the two halves would match perfectly.
Understanding these properties in the sunburst window allows for more effective architectural and design applications, particularly when handling light and space.

Ellipse Standard Equation

The ellipse standard equation is a formula that expresses every point on the ellipse in terms of its coordinates. When an ellipse is centered at the origin, its standard equation can be written as:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Here, \( a \) is the semi-major axis and \( b \) is the semi-minor axis. This equation helps us calculate any point on the ellipse by substituting the desired \( x \) or \( y \) values.
In solving the sunburst window problem, this form allows us to substitute known values to find unknowns. For example, when \( x = 25 \), plug it into the equation to solve for \( y \):
\[ \frac{25^2}{1600} + \frac{y^2}{400} = 1 \]
By isolating and solving for \( y \), we find the height of the ellipse at a specific point. This practicality makes the standard equation of the ellipse a crucial tool for solving geometric problems.