Q. 115 - Precalculus Enhanced with Graphing Utilities

Question

Approximating the Circumference of Earth Eratosthenes of Cyrene (276–195 bc) was a Greek scholar who lived and worked in Cyrene and Alexandria. One day while visiting in

Syene he noticed that the Sun’s rays shone directly down a well. On this date 1 year later, in Alexandria, which is 500 miles due north of Syene he measured the angle of the Sun to be about 6.2 degrees. See the figure. Use this information to approximate the radius and circumference of Earth.

Step-by-Step Solution

Answer

The approximate value of the circumference is 25000 mi.

Step 1: Set Up Proportion

Arc of 500 miles corresponds to central angle of about \(7.2°\) (using the problem's approximation).

Step 2: Calculate

\(C = \frac{500 \times 360}{7.2} = 25{,}000\) miles. Radius \(r = C/(2\pi) \approx 3{,}979\) miles.