Project #4 Exponential Growth Models

MATH 1113 PreCalculus

Day 34, Project #5
Another Maximum Angle Problem

This one will be a bit easier than Project #7 (I hope), and I added it after looking at problem #95 on page 565. You won't be asked to find the formula (although you certainly should be able to), but I do want you to explore the very interesting relationship between the heights of the buildings, the distance between them, and the placement that maximizes the angle. Just to be original, I've changed the problem from a shrub to a telescope.

Maximizing the viewing angle

A telescope is to be placed between two buildings at a location which maximizes the viewing angle for the telescope.

Here's the Geogebra applet to help you explore the situation:

Applet:

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, right click to zoom out and back in. This will clear the screen ("control-F" doesn't work on an embedded applet.)

Assignment:

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?

In a Calculus class, you'll see how to prove answers to problems such as these (in fact, you'll need multi-variable Calculus to answer questions like #2.)

Page Last Modified: 21 March, 2007

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