USAAAO 2024, I кръг — Задача 29
USAAAO 2024 - First Round · 10 февруари 2024 г.
Problem writers
Wesley Andrade, Abhay Bestrapalli, Austin Chen, Justin Chen, Ved Gund, Hagan Hensley, Sandesh Kalantre, Evan Kim, David Lee, Joe McCarty, Arjun Patel, Sahil Pontula, Tara Rezaei, Raul Teixeira, Leo Yao, David Zhang
Условие
After that slight headache, Austin is back at MIT in Boston! For his astronomy research, he is observing the LARES satellite which is a ball of diameter 36.4 cm made out of THA-18N (a tungsten alloy). It orbits at a distance 1450 km from the surface of the Earth and at an inclination of relative to the equatorial plane. What is the highest altitude Austin can point his telescope if he wants to observe LARES at its highest latitude?
(a)
(b)
(c)
(d)
(e)
Отговори
- A
Решение
Solution:
Let the radius of the Earth be , the distance of LARES from the surface of the Earth be , LARES’ inclination be , latitude of Boston be , and the distance of LARES from Boston be . Consider the triangle with vertices of the center of the Earth, Boston, and LARES’ maximal latitude position. (This triangle lies in the plane containing the North and South poles). Then from Law of Cosines
Let the corresponding altitude be . Then from Law of Sines
Answer: A
Оригинал в Архива: usaaao_first_exam_2024-3.pdf · официални решения: usaaao_first_exam_2024_solutions-1.pdf