Chapter 7.3, Problem 110AE

Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length $L$ of the ladder as a function of $\theta$ is $L\left(\theta \right)=4\csc \theta +3\sec \theta$.

a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation $3\sec \theta \tan \theta -4\csc \theta \cot \theta =0$ $0^{\circ }<\theta <{90}^{\circ }$ Solve this equation for $\theta$.

b. What is the length of the longest ladder that can be carried around the corner?

c. Graph $L=L\left(\theta \right)$, $0^{\circ }\le \theta \le {90}^{\circ }$, and find the angle $\theta$ that minimizes the length $L$.

d. Compare the result with the one found in part (a). Explain why the two answers are the same.