Chapter 4.6, Problem 33E

To determine

To calculate: The rate at which the shadow length decreasing

Answer to Problem 33E

Shadow length is decreasing by $7.1\text{in}/\text{min}$

Explanation of Solution

Given information:

The shadow of a $80ft$ building on level ground is $60ft$ long.

The angle $\theta$ the sun makes with the ground is increasing at the rate of ${0.27}^{°}/\text{min}$ .

Calculation:

Organize the given information. Multiply by $\frac{\pi }{{180}^{°}}$ to convert to radians.

  $\begin{array}{c}\frac{d\theta }{dt}={0.27}^{°}/\text{min}\\ =0.00471radians/\min \end{array}$

Let $s$ represent the length of the shadow.

At the moment, $s=60\text{ft}$ , so

  $\theta =\arctan (80/60)\approx {53.13}^{°}$

Find $\frac{ds}{dt}$ , the rate at which $s$ is decreasing.

Use trigonometry to get an equation relating $\theta$ and $s$ . Implicitly differentiate with respect to time $t$ .

  $\begin{array}{c}\tan \theta =\frac{80}{s}\\ s=\frac{80}{\tan \theta }\\ =80\cot \theta \end{array}$

Plug in the known values to get $\frac{ds}{dt}$ .

Use radians for $\frac{d\theta }{dt}$ because when making calculations of an angle value and another type of value, degrees don't work.

  $\begin{array}{c}\frac{ds}{dt}=80\left(-{\csc }^2\theta \right)\frac{d\theta }{dt}\\ =-80\left({\csc }^2{53.13}^{°}\right)(0.00471)\\ \approx -0.58875\text{ft}/\text{min}\end{array}$

The answer is in $\text{ft}/\text{min}$ , so multiply by $12$ to get $in/\min$ . It is negative, which indicates $s$ is decreasing.

  $\approx -7.065\text{in}/\text{min}$