# Thread: Finding a rate of change, and mean value theorem

1. ## Finding a rate of change, and mean value theorem

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building.

And

Find numbers that satisfy the conclusion of the mean value theorem for f(x) = x/(x-1) on [2, 5]

Thanks to anyone that answers the post!

2. Originally Posted by Gfreak
A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building.

And

Find numbers that satisfy the conclusion of the mean value theorem for f(x) = x/(x-1) on [2, 5]

Thanks to anyone that answers the post!
For the first one use similar triangles.

Let y be the height of the shadow on the wall and let x be the distance from the light then

$\frac{y}{2}=\frac{12}{x} \implies \frac{dy}{dt}=-\frac{24}{x^2}\frac{dx}{dt}$

For the 2nd one just solve the equaiton.

$\frac{f(5)-f(2)}{5-2}=f'(c)$ for c.