Multiple choice

A man running round a race course notes that the sum of the distance of two flag posts from him is always $10\ m$ and the distance between the flag posts is $8\ m$. The area of the path he encloses in square metres is:

  1. $15 \pi$
  2. $20 \pi$
  3. $27 \pi$
  4. $30 \pi$
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A Correct answer
Explanation

The path is an ellipse where the sum of distances to foci is 2a = 10 (a=5) and distance between foci is 2ae = 8 (ae=4). Thus e = 0.8. b^2 = a^2(1-e^2) = 25(1-0.64) = 25(0.36) = 9. Area = pi * a * b = pi * 5 * 3 = 15 pi.

AI explanation

The constant sum of distances from two fixed points defines an ellipse, where the major axis length 2a equals 10 meters, making a equal to 5 meters. The distance between the flag posts represents the focal length 2c, which is 8 meters, so c equals 4 meters. Using the ellipse relationship b^2 = a^2 - c^2, we find b^2 = 5^2 - 4^2, which equals 25 minus 16, resulting in 9. The area of the elliptical path is pi*a*b, which calculates to 15*pi square meters.

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