Final answer:
By using a proportion based on the angle corresponding to the Tropic of Cancer, we can calculate the Earth's circumference to be approximately 39,940.4 km.
Step-by-step explanation:
The student is asking about the calculation of the Earth's circumference based on observations made at the Equator and the Tropic of Cancer, which involves the concepts of shadow casting, solar declination, and latitudinal degrees.
On a specific day, a vertical stick at the Tropic of Cancer (23.5° N latitude) casts a shadow while one on the Equator does not, indicating that the Sun is directly overhead at the Equator. By understanding that 1 degree of latitude is approximately 111 km, we can calculate the circumference of the Earth. Here's a step-by-step explanation:
- At the Equator, the Sun is directly overhead, so there is no shadow. This means the solar rays are hitting the Equator at a 90° angle.
- At the Tropic of Cancer, the stick casts a shadow. This means the solar rays are at an angle at this latitude.
- The difference in latitude between the Equator and the Tropic of Cancer is 23.5°. Since this is the angle of the Sun's rays at the Tropic of Cancer, we use this angle to calculate the circumference.
- Knowing that one degree of latitude is approximately 111 km, 23.5° equals 23.5 × 111 km = 2608.5 km.
- The 23.5° angle represents the arc length of 2608.5 km, which is part of the full 360° circle of the Earth's circumference. Therefore, we can calculate the full circumference (C) using the proportion 23.5°/360° = 2608.5 km/C.
- Solving for C, we find that C = (360 × 2608.5 km) / 23.5 = 39,940.4 km.
Therefore, the Earth's circumference is approximately 39,940.4 km.