1. ## Rate of Change

Simple calculus question that i just cannot get my brain around currently.

A lamp is 6m directly above a straight path. A man 2m tall walks along the path away from the light at a constant speed of 1 m/s. At what speed is the end of his shadow moving along the path? At what speed is the length of his shadow increasing.

A full working would be appreciated. But even if just show me the set up that would still be good.

2. Originally Posted by lukybear
simple calculus question that i just cannot get my brain around currently.

A lamp is 6m directly above a straight path. A man 2m tall walks along the path away from the light at a constant speed of 1 m/s. At what speed is the end of his shadow moving along the path?

Let $\displaystyle r=x+s$

We are given that $\displaystyle \frac{dx}{dt}=1$ and we want to find $\displaystyle \frac{dr}{dt}$

$\displaystyle \frac{dr}{dt}=\frac{dx}{dt}+\frac{ds}{dt}$

The sides of the similar triangles are proportional so

$\displaystyle \frac{x+s}{s}=\frac{6}{2}=3$

$\displaystyle x+s=3s$

$\displaystyle x=2s$

$\displaystyle \frac{dx}{dt}=2\frac{ds}{dt}$

$\displaystyle \frac{ds}{dt}=\frac{1}{2}\frac{dx}{dt}$

$\displaystyle \frac{dr}{dt}=\frac{dx}{dt}+\frac{1}{2}\frac{dx}{d t}=1.5 \frac{m}{s}$

Originally Posted by lukybear
at what speed is the length of his shadow increasing.

A full working would be appreciated. But even if just show me the set up that would still be good.
We want to find $\displaystyle \frac{ds}{dt}$