1. ## rates problem

Can anyone help with this rates problem?

A light is on the ground, 15 m from a wall. A woman, 1.6 m tall, walks away from the light and towards the wall at 0.5 m/s. Calculate the rate of change of the length of her shadow on the wall when the woman is 10 m from the wall.

I guess you would have to find the rate at which the tip of her shadow is moving? (I think)

2. Hello, math619!

A light is on the ground, 15 m from a wall.
A woman, 1.6 m tall, walks away from the light and towards the wall at 0.5 m/s.
Calculate the rate of change of the length of her shadow on the wall
when the woman is 10 m from the wall.

I guess you would have to find the rate at which the tip of her shadow is moving.
. . Right!
Code:
                              *
*  |
*     |
*        |
*           | s
*  |           |
*     |1.6        |
*   x    |           |
*-----------+-----------*
: - - - -  15 - - - - - :

From similar right triangles, we have: .$\displaystyle \frac{s}{15} \:=\:\frac{1.6}{x}$
. . and we have: .$\displaystyle s \:=\:24x^{-1}$

Differentiate with respect to time: .$\displaystyle \frac{ds}{dt}\:=\:-\frac{24}{x^2}\ \frac{dx}{dt}$

We are given: .$\displaystyle \frac{dx}{dt} = 0.5$ . . . and "10 m from the wall" means $\displaystyle x = 5$

. . Therefore: .$\displaystyle \frac{ds}{dt} \:=\:-\frac{24}{5^2}(0.5) \;=\;\boxed{-0.48\text{ m/sec}}$