Problem:
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is from Alice's position and from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let Alice, Bob, and the airplane be located at points , and , respectively. Let be the point on the ground directly beneath the airplane, and let be the airplane's altitude, in miles. Then and are right triangles with right angles at , so and . Then by the Pythagorean Theorem applied to the right triangle on the ground,
Thus , and the closest of the given choices is .
The problems on this page are the property of the MAA's American Mathematics Competitions