Maximizing the viewing angle

A telescope is to be placed between two buildings at a location which maximizes the viewing angle for the telescope. Adjust the heights of the two buildings and the distance between the buildings. Then slide the telescope location back and forth and observe the traced curve, representing viewing angle. If you change the heights of the buildings or the distance between them, press "Control+F" to clear the screen.

1. Set the first building's height at 200 ft, the second one's at 120 ft, and the distance between them at 240 ft. Find the location that maximizes the angle.
A) What is the location?
B) What is the angle?

2. For what relationships between the heights of the buildings, and the distance between them, is the maximum angle achieved right at the edge of the taller building? Report on several different scenarios:

A) Suppose that the gray building is twice as tall as the brown one. How far apart/close together do they have to be before you're best off putting the telescope right against the gray building? (Try 250 feet for the gray building and 125 for the brown building.)

B) Suppose that the gray building is three times as tall as the brown one. How far apart/close together do they have to be before you're best off putting the telescope right against the gray building? (Try 300 feet for the gray building and 100 for the brown one.)

C) Suppose that the distance between the buildings is 3 times the height of the tan building. How tall must the gray building be before the best location is right against it? Try, for example, a height for the tan building of 32 feet, and a distance apart of 96 feet.

3. For what relationships between the heights of the buildings, and the distance between them, is the maximum angle achieved in the middle between the buildings?

Dave Matthews, April 2, 2007, Created with GeoGebra