Question

You are an astronomer on planet Tirth, which orbits a distant star. It has recently been accepted that Tirth is spherical in shape, though no one knows its size. One day, you learn that on the equinox your sun is directly overhead in the city of Tyene, located 350 kilometers due north of you. On the equinox, you go outside in Alectown and observe that the altitude of your sun is 85

a. What is the circumference of Tirth?

Answer 1

To calculate the circumference of Tirth, you can use the principle that Eratosthenes used to estimate the size of Earth. The altitude of the sun in Alectown on the equinox is 85 degrees, and the sun is directly overhead in Tyene. By using the angle difference and the distance between the two locations, you can calculate the circumference of Tirth to be 25,200 kilometers.

Step-by-step explanation:

To calculate the circumference of Tirth, we can use the same principle that Eratosthenes used to estimate the size of Earth. On the equinox, the sun is directly overhead in Tyene, which is located 350 kilometers north of Alectown. In Alectown, the altitude of the sun is 85 degrees. We can calculate the circumference of Tirth by using the angle difference between the two locations and the distance between them.

The angle difference between the two locations is given by 90 degrees minus the altitude of the sun in Alectown, which is 85 degrees. So the angle difference is 90 - 85 = 5 degrees.

Using the principle that the angle difference is proportional to the distance on a spherical surface, we can calculate the circumference of Tirth. We can set up the following proportion:

Angle difference in degrees / 360 degrees = Distance between the two locations / Circumference of Tirth

Substituting the values, we get:

5 degrees / 360 degrees = 350 kilometers / Circumference of Tirth

Cross-multiplying and solving for the circumference, we get:

Circumference of Tirth = (360 degrees * 350 kilometers) / 5 degrees = 25,200 kilometers.