Chapter 6.5, Problem 51AE

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.

(a) Show that the length $L$ of the ladder shown as a function of the angle $\theta$ is

$L\left(\theta \right)=3\sec \theta +4\csc \theta$

(b) Graph $L=L\left(\theta \right),\text{0}\text{<}\theta \text{<}\frac{\pi }{2}$

(c) For what value of $\theta$ is $L$ the least?

(d) What is the length of the longest ladder that can be carried around the corner? Why is this also the least value of $L$?