Question

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The base of an auditorium is in the form of an ellipse 200 ft. long and 100 ft. wide. A pin dropped near one focus can clearly be heard at the other focus. Determine the distance between the foci to the nearest tenth of a foot. Show each step of the solution.

Answer 1

Step 1 — Identify semi-axes:
- The major axis length = 200 ft, so a = 200/2 = 100 ft.
- The minor axis length = 100 ft, so b = 100/2 = 50 ft.

Step 2 — Use the focal distance formula for an ellipse:
- For an ellipse, c^2 = a^2 − b^2, where 2c is the distance between the foci.

Compute c:
c = sqrt(a^2 − b^2) = sqrt(100^2 − 50^2) = sqrt(10000 − 2500) = sqrt(7500).

Simplify/approximate:
sqrt(7500) = 10·sqrt(75) ≈ 86.6025403784 ft.

Step 3 — Distance between foci = 2c:
2c ≈ 2 · 86.6025403784 ≈ 173.2050807568 ft.

Round to nearest tenth:
Distance ≈ 173.2 ft.

Answer: The distance between the foci is approximately 173.2 feet.