Question

A fireplace arch is to be constructed in the form of a semiellipse. The opening is to

have a height of 1 foot at the center and a width of 6 feet along the base. The
contractor cuts a string of a certain length and nails each end of the string at the foci
of the base in order to sketch the outline of the semiellipse.
a.) How far from the center should the string be nailed into the base?

Answer 1

2√2 ft ≈ 2.83 ft ≈ 2 ft 9.9 in

Explanation:

You want to know the distance of the foci from the center of an ellipse that is 6 ft long and has semi-minor axis of 1 ft.

Center distance

The distance 'c' from the center of the ellipse is related to the semi-major axis 'a' and the semi-minor axis 'b' by ...

a² = b² +c²

Solving for c, we have ...

c = √(a² -b²)

The 6 ft length of the ellipse tells us a=3. The 1 ft height from the midline of the ellipse tells us b=1. This gives c as ...

c = √(3² -1²) = √8 = 2√2 . . . . . feet

The string should be nailed 2√2 feet, about 2.83 feet or 2 ft 9.9 inches from the center of the ellipse.

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Additional comment

The foci are at a distance of 3 ft from the top center of the ellipse, so the 6 ft string could be used to find the foci directly. The contractor can measure 1 ft up from the center of the ellipse, and 3 ft from that point back to the base of the arch. The string used to outline the ellipse will be 6 ft long, the length of the major axis of the ellipse.

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