# Rate of change

• Nov 3rd 2010, 05:43 PM
Johnson
Rate of change
Gravel is being dumped from a conveyor belt at a
rate of 40 ft^3/min. It forms a pile in the shape of a right circular
cone whose base diameter and height are always the same.
How fast is the height of the pile increasing when the pile is
18 ft high?
The height is increasing at ?? ft/min?
• Nov 3rd 2010, 05:52 PM
skeeter
Quote:

Originally Posted by Johnson
Gravel is being dumped from a conveyor belt at a
rate of 40 ft^3/min. It forms a pile in the shape of a right circular
cone whose base diameter and height are always the same.
How fast is the height of the pile increasing when the pile is
18 ft high?
The height is increasing at ?? ft/min?

you are given ...

$\displaystyle \frac{dV}{dt} = 40 \, ft^3/min$

$\displaystyle 2r = h$

gravel pile is in the form of a cone ... $\displaystyle V = \frac{\pi}{3} r^2 h$

the problem wants you to find $\displaystyle \frac{dh}{dt}$ when $\displaystyle h = 18 \, ft$

get the volume in terms of $\displaystyle h$ , take the derivative w/r to time, and substitute your given values. finally, solve for $\displaystyle \frac{dh}{dt}$.