Chapter 4.1, Problem 126PE

In the third century $\text{B}\text{.C}\text{.}$ , the Greek astronomer Eratosthenes approximated the Earth's circumference. On the summer solstice at noon in Alexandria, Egypt, Eratosthenes measured the angle $\alpha$ of the Sun relative to a line perpendicular to the ground. At the same time in Siena (now Aswan), located on the Tropic of Cancer, the Sun was directly overhead.

a. If $\alpha =\frac{1}{50}$ of a circle, find the measure of $\alpha$ in degrees. (In Eratosthenes' time, the degree measure had not yet been defined.)

b. If the distance between Alexandria and Siena is $5000$ stadia, find the circumference of the Earth measured in stadia.

c. If $10\text{ stadia}\approx 1\text{ mi}$ , find Eratosthenes' approximation of the circumference of the Earth in miles (the modern-day approximation at the equator is $24,900\text{ mi}$ ).