Q. 46 - Calculus

Question

In Exercises 46–48, suppose that Stuart is 6 feet tall and is walking towards a 20-foot streetlight at a rate of 4 feet per second. As he walks towards the streetlight, his shadow gets shorter. 

How fast is the length of Stuart’s shadow changing? Does it depend on how far Stuart is from the streetlight? 

Step-by-Step Solution

Answer

1.71 feet per second at constant rate.

Step 1. Given information

Given is :

suppose that Stuart is 6 feet tall and is walking towards a 20-foot streetlight at a rate of 4 feet per second. As he walks towards the streetlight, his shadow gets shorter.  

We have to answer that how fast is the length of Stuart’s shadow changing.

Step 2. Length of Stuart’s shadow changing.

We are given the rate at which Stuart walks towards the streetlight, and we wish to find the rate of change of the length of Stuart's shadow. To find a relationship between these two rates we will find a relationship between their underlying variables: the distance s between Stuart and the streetlight and the length l of Stuart's shadow.

So,

$\begin{array}{r}\frac{20}{s+l}=\frac{6}{l}\\ 20l=6s+6l\\ 14l=6s\\ 7l=3s\end{array}$

Differentiate with respect to time,

$\begin{array}{r}\frac{dl}{dt}=\frac{3}{7}\frac{ds}{dt}\\ =\frac{3}{7}\times 4\\ =\frac{12}{7}\end{array}$