A small airplane at an altitude of 5000 feet is flying east at 300 feet per second, and you are watching it with a small telescope as it passes directly overhead.

Question

A) what is the slope of the telescope 5, 10, and 20 seconds after tge plane passes overhead? B)what is the slope of the telescope t seconds after the plane passes overhead ?

Answer 1

In t seconds the plane moves 300t feet further east. The plane was initially at an altitude of 5000 feet. What we have is a right triangle with the long leg=5000 and the short leg=300t. As the plane moves the telescope is tilted further and further away from the vertical. The slope (tangent) of this angle is 300t/5000=3t/50.

(A) Now we can put t=5, 10 and 20: 0.3, 0.6, 1.2. The slopes relative to the ground are the inverses of these:

10/3, 5/3, 5/6. The corresponding angles are 73.30, 59.04, 39.81 degrees. The angles from the vertical are these angles subtracted from 90 degrees: 16.70, 30.96, 50.19 degrees.

(B) The slope (from the ground, or angle of tilt of the telescope) is 50/3t or arctan(50/3t) (same as tan^-1(50/3t)).