Problem 54 - Algebra and Trigonometry

Question

Plywood Ellipse A carpenter wishes to construct an elliptical table top from a sheet of plywood, 4 ft by 8 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; place tacks 4√3 feet apart.

Step 1: Identify the Dimensions of the Plywood Sheet

The plywood sheet measures 4 feet by 8 feet. This means the maximum possible minor diameter (width of the ellipse) is 4 feet and the maximum major diameter (length of the ellipse) is 8 feet. These dimensions will form the semi-minor axis \(b = 2\) feet and the semi-major axis \(a = 4\) feet of the ellipse.

Step 2: Understand the Thumbtack and String Method

The 'thumbtack and string' method is used for drawing an ellipse. Two thumbtacks are placed at the ellipse's foci, and a string is looped around them, creating a triangle with the pencil. The total length of this string equals the ellipse's major axis length \(2a\). The sum of the distances from any point in the ellipse to the two foci (thumbtacks) is constant and equals \(2a\).

Step 3: Calculate the Distance of the Foci

The distance \(c\) between the center of the ellipse and each focus is given by the equation \(c = \sqrt{a^2 - b^2}\). Here, \(a = 4\) and \(b = 2\), so:\[c = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3}\]

Step 4: Determine String Length and Tack Spacing

The length of the string that forms the ellipse is equal to the major axis length \(2a = 8\) feet. The thumbtacks (foci) need to be placed \(2c\) feet apart. Since \(c = 2\sqrt{3}\), the spacing between the thumbtacks is:\[2c = 2 \times 2\sqrt{3} = 4\sqrt{3}\,\text{feet}\]This is the maximum spacing and string length for the largest possible ellipse within the constraints of the plywood dimensions.

Key Concepts

Ellipse

An ellipse is a fascinating shape that belongs to the category of conic sections, formed when a plane cuts through a cone at an angle that is not perpendicular to the cone's base. It appears like a stretched-out circle and has two axes – the major and minor axes. The points on the ellipse are such that the sum of the distances to two fixed points, called foci, is constant. This unique property gives the ellipse its distinct shape. Ellipses have many real-world applications, ranging from planetary orbits to architectural designs. Understanding the geometry of an ellipse can help you appreciate these applications and better visualize them.

Thumbtack and String Method

The thumbtack and string method is a simple, practical technique for drawing an ellipse. Here's how it works:

• Two thumbtacks are placed on a surface at the foci of the desired ellipse.
• A piece of string is tied in a loop around the tacks.
• A pencil is used to pull the string taut while all three points remain on the surface.

This forms a triangle with the string around the pencil tip, creating a locus of points that trace out an ellipse as you move the pencil around. The key here is that the length of the string keeps a constant distance, equal to the ellipse’s major axis, ensuring that the ellipse is perfectly formed.Whether you're crafting an elliptical table or just exploring geometry, this method offers a practical and visual way to understand and create ellipses.

Semi-Major Axis

The semi-major axis is one of the two main measures of an ellipse, depicting the longest radius from the center to the perimeter. For any given ellipse, the major axis has a length of two times the semi-major axis, stretching across the entire figure.

An ellipse’s size and shape heavily depend on the length of this semi-major axis. The longer it is, the more stretched out the ellipse appears. In the context of the table problem, the semi-major axis was determined to be 4 feet, which is half the longer side of the plywood. This measurement directly affects how the ellipse fits into the available material while maintaining its proportions.

Semi-Minor Axis

The semi-minor axis is the other key dimension of an ellipse. It represents the shorter radius, running perpendicular to the semi-major axis at the ellipse's widest point. Like the semi-major axis, the length of the semi-minor axis is critical in defining the ellipse's shape and size.

The shorter this radius, the more "squashed" the ellipse looks. In the carpenter's project for an elliptical table, the semi-minor axis was 2 feet long, which is half the width of the plywood. Understanding the balance between the semi-major and semi-minor axes helps in comprehending how an ellipse fits differently shaped spaces efficiently.