Problem 51 - 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 high- est 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 is approximately 15.61 inches.

Step 1: Setup the Ellipse Equation

Considering the window as the top half of an ellipse centered at the origin, we begin by using the standard ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

Step 2: Determine Semi-major and Semi-minor Axes

Since the window is 80 in. wide, the total major axis is 80 in., meaning \(a = \frac{80}{2} = 40\) in. The highest point of 20 in. represents the semi-minor axis, so \(b = 20\) in.

Step 3: Plug Values into the Equation

Substituting \(a = 40\) and \(b = 20\) into the ellipse equation gives us \(\frac{x^2}{40^2} + \frac{y^2}{20^2} = 1\). As the sunburst window is the top half, \(y\) must be non-negative.

Step 4: Find the Height 25 in. from Center

Substitute \(x = 25\) into the equation \(\frac{25^2}{40^2} + \frac{y^2}{20^2} = 1\). This simplifies to \(\frac{625}{1600} + \frac{y^2}{400} = 1\).

Step 5: Solve for y

First calculate \(\frac{625}{1600} = 0.390625\). Hence, \(\frac{y^2}{400} = 1 - 0.390625 = 0.609375\). Multiply through by 400 to get \(y^2 = 400 \times 0.609375 = 243.75\). Taking the square root gives \(y = \sqrt{243.75} \approx 15.61\) in.

Key Concepts

Semi-major Axis

In any ellipse, the **semi-major axis** is one of the most important aspects to understand. It is essentially half of the longest diameter of the ellipse. This axis runs through the center from one side to the other. In the context of our problem, the sunburst window is 80 inches wide at the bottom.
However, since the ellipse is symmetric with respect to its center, the semi-major axis is half of that width, which equals 40 inches. The semi-major axis helps define the overall "stretch" or "elongation" of the ellipse.
Understanding its length is crucial in forming the ellipse's equation, which graphically represents the window's shape. Whenever you deal with an ellipse, knowing the semi-major axis gives you a starting point for determining other axis-related properties.

Semi-minor Axis

Parallel to understanding the semi-major axis, the **semi-minor axis** describes half of the shortest diameter passing through the ellipse's center.
For the window, the tallest point, 20 inches from the base, stands for the semi-minor axis. The semi-minor axis is always perpendicular to the semi-major axis. Together, they give a clear idea of the shape of the ellipse and are vital when constructing the ellipse equation.
This minor axis provides information on the height the ellipse achieves at its zenith and defines the "squash" of the elliptical shape. Knowing both axes is key to mastering how an ellipse fits into given physical constraints or design specifications.

Ellipse Equation

The standard **ellipse equation** plays a crucial role in defining the shape mathematically. For an ellipse centered at the origin, the equation 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.
This equation is integral to solving problems involving ellipses, such as determining points on the ellipse given certain constraints.In our sunburst window scenario, we substitute \( a = 40 \) and \( b = 20 \) to get the specific ellipse equation \( \frac{x^2}{40^2} + \frac{y^2}{20^2} = 1 \).
This equation can handle many queries, including finding the height of the window at specific distances from the center. Once values are plugged in for \( x \), solving for \( y \) (as with our height calculation at 25 inches from the center) unveils the corresponding point on the window's shape.
Understanding how to use and manipulate this equation will aid significantly in interpreting the properties and dimensions of the ellipse effectively.