Question

John is making an elliptical shaped table from a rectangular piece of wood. He puts 2 nails in the wood 16 inches apart and then attaches 20 inches of string between the nails. Using a pencil, he pulls the string tight and thus traces out the set of...?

all points whose sum of distances to the 2 nails is 20 inches, which is an ellipse. In inches, what is the sum of the lengths of the major and minor axes of the ellipse?

Answer 1

Major Axis `= 20`
Minor Axis `= 12`
SUM `= 32`

Explanation:

Nice descriptions of a practical ellipse generation! The equation for an ellipse is not necessary for this problem because it is only looking for the major and minor axis lengths. The major axis is simply the length between the nails (16) plus the difference with the string length, or the original string length.

The minor axis is twice the height of the isosceles triangle formed when the string is extended at the midpoint of the nails.

Major Axis `= 16 + 4 = 20`

The triangle has sides of 10, 10 and 16. So the side angles are:
`cos(phi) = 8/10 = 0.8`; `phi = 36.9.^o`
Height: `tan(36.9) = h/8` ; `h = 6`
Minor Axis `= 2 xx h = 2 xx 6 = 12`
SUM `= 32`