Chapter 7.3, Problem 108AYU

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.

To determine

To find:

a. In calculus, you will be asked to find the length of the longest ladder that can turn the corner by solving the equation.

$3\sec \theta \tan \theta -4\csc \theta \cot \theta =0,0°<\theta <90°$ Solve this equation for $\theta$.

Answer to Problem 108AYU

a. $\tan \theta =\sqrt[3]{\frac{4}{3}}$

Explanation of Solution

Given:

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,

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

Calculation:

a. $3\sec \theta \tan \theta -4\csc \theta \cot \theta =0$

$\frac{3}{\cos \theta }\frac{\sin \theta }{\cos \theta }-\frac{4}{\sin \theta }\frac{\cos \theta }{\sin \theta }=0$

$\frac{3\sin \theta }{{\cos }^2\theta }-\frac{4\cos \theta }{{\sin }^2\theta }=0$

Multiply by ${\cos }^2\theta {\sin }^2\theta$

$3{\sin }^3\theta -4{\cos }^3\theta =0$

$3{\sin }^3\theta =4{\cos }^3\theta$

${\tan }^3\theta =\frac{4}{3}$

$\tan \theta =\sqrt[3]{\frac{4}{3}}$

To determine

To find:

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

Answer to Problem 108AYU

b. $9.8656$ feet.

Explanation of Solution

Given:

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,

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

Calculation:

b. The maximum happens when $x={\tan }^{-1}\left(\sqrt[3]{\frac{4}{3}}\right)$ sub that back into the original equation to get the ladder length. Here is where you would use your calculator to get decimal approximations.

$x={\tan }^{-1}\left(\sqrt[3]{\frac{4}{3}}\right)$

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

$L\left(x\right)=4\csc \left({\tan }^{-1}\left(\sqrt[3]{\frac{4}{3}}\right)\right)+3\sec \left({\tan }^{-1}\left(\sqrt[3]{\frac{4}{3}}\right)\right)$

Without showing a crap-load of work, I can conclude that the maximum amount of "ladder" than can fit is exactly $7\surd \left(³\surd 3 +³\surd 4\right)/³\surd 4$, which is approximately $\text{9}\text{.8656}$ feet.

To determine

To find:

c. Graph $L=L\left(\theta \right),0°\le \theta \le 90°$ and find the angle $\theta$ that minimizes the length $l$ .

Answer to Problem 108AYU

c.

Explanation of Solution

Given:

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,

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

Calculation:

c.

Minimum length is $\text{9}\text{.86}$ feet

To determine

To find:

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

Answer to Problem 108AYU

d. Two answers are same, since both are angle of inclination.

Explanation of Solution

Given:

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,

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

Calculation:

d. Two answers are same, since both are angle of inclination.