Problem 50 - Algebra and Trigonometry

Question

Plywood Ellipse A carpenter wishes to construct an elliptical table top from a sheet of plywood, 4 \(\mathrm{ft}\) by 8 \(\mathrm{ft}\) . He will trace out the ellipse using the "thumbtack and string" method illustrated in Figures 2 and \(3 .\) What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?

Step-by-Step Solution

Answer

Use an 8-foot string, and place tacks 4\(\sqrt{3}\) feet apart.

Step 1: Identify the Semi-major and Semi-minor Axes

The largest ellipse that can be inscribed in a rectangle will have its semi-major axis along the larger dimension and its semi-minor axis along the smaller dimension of the rectangle. For the plywood, the dimensions are 4 feet by 8 feet, so the semi-major axis \(a\) is 4 feet (half of 8 feet), and the semi-minor axis \(b\) is 2 feet (half of 4 feet).

Step 2: Determine the String Length Needed

In the 'thumbtack and string' method, the length of the string should equal the sum of the distances from the two foci to any point on the ellipse, which equals \(2a\). For our ellipse, the string should be twice the semi-major axis, so it is \(2 \times 4 = 8\) feet long.

Step 3: Calculate the Distance between the Foci

The sum of the distances from the two foci to any point on the ellipse is constant (the string length). This distance \(2c\) between the foci for an ellipse is found by the equation \(c = \sqrt{a^2 - b^2}\). Substituting the values: \(c = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3}\). Hence, the distance between the two tacks should be \(2c = 4\sqrt{3}\) feet.

Key Concepts

Semi-major Axis

The semi-major axis of an ellipse is one of its most defining features. It is the longest radius of the ellipse and spans the largest distance from the center to the edge. For any ellipse inscribed within a rectangle, the semi-major axis aligns with the rectangle's longest dimension. In our carpenter's task, the plywood is 8 feet across its longest side. The semi-major axis, therefore, is half of this length, making it 4 feet. This distance dictates the overall size and "stretch" of the ellipse, giving it that elongated look.

Semi-minor Axis

The semi-minor axis is the shortest radius of an ellipse and runs perpendicular to the semi-major axis. It measures from the center to the edge of the ellipse through its shortest span. For the plywood table project, the smaller side of the sheet is 4 feet. Therefore, the semi-minor axis is half of this distance, so it measures 2 feet. This axis provides depth to the ellipse, counterbalancing the elongation from the semi-major axis, and defines how "narrow" the ellipse appears.

Thumbtack and String Method

The thumbtack and string method is a classic technique for tracing an accurate ellipse. Here's how it works:

• Insert two thumbtacks at the foci of the desired ellipse on your plywood.
• Use a string that is the total length of 8 feet (equal to twice the semi-major axis in this task).
• Attach the string's ends to the thumbtacks, creating a loop.
• Pull the string taut with a pencil and trace a curve as you move the pencil around, keeping the string tight.

This method relies on the principle that the sum of distances from any point on the ellipse to the foci is constant, facilitating a smooth and accurate ellipse outline.

Foci of an Ellipse

The foci (singular: focus) are two important fixed points inside an ellipse. The unique property of an ellipse is that the total distance from these two points to any point on the ellipse's boundary is always the same, which is crucial for drawing it accurately. In our elliptical table scenario, find the foci using the formula:\[ c = \sqrt{a^2 - b^2} \]Substituting our known values, where \(a = 4\) feet and \(b = 2\) feet, gives us:\[ c = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \]Therefore, the foci are spaced \(2c = 4\sqrt{3}\) feet apart. This precise measure allows for the proper application of the thumbtack and string method to ensure an even, symmetrical ellipse.