# rates of change

• Oct 29th 2007, 08:59 PM
mer1988
rates of change
Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of the cone.

(a) Find the rate at which the radius of the cone increases when the radius is 2 meters.
dr/dt = meters per hour

(b) Find the rate at which the height of the cone increases when the radius is 2 meters.
dh/dt = meters per hour

and

A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.08 meter. (a) At that moment, how fast is the area of the slick expanding?

(b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing when the radius is 150 meters?
• Oct 29th 2007, 11:12 PM
kalagota
Quote:

Originally Posted by mer1988
Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of the cone.

(a) Find the rate at which the radius of the cone increases when the radius is 2 meters.
dr/dt = meters per hour

(b) Find the rate at which the height of the cone increases when the radius is 2 meters.
dh/dt = meters per hour

a)
$\displaystyle V_{cone}=\frac{1}{3} \pi r^2 h$
note that $\displaystyle h=2r$ and $\displaystyle \frac{dV}{dt}=0.6 m^3/hr$
so
$\displaystyle V_{cone}=\frac{1}{3} \pi r^2 h=\frac{2}{3}\pi r^3$
$\displaystyle \frac{dV}{dt}=\frac{6}{3}\pi r^2 \frac{dr}{dt}$ then evaluate at r=2 to solve for $\displaystyle \frac{dr}{dt}$

b)
$\displaystyle h=2r$, then
$\displaystyle \frac{dh}{dt}=2\frac{dr}{dt}$

Quote:

Originally Posted by mer1988
and
A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.08 meter.
(a) At that moment, how fast is the area of the slick expanding?

(b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing when the radius is 150 meters?

a)
$\displaystyle A_{slick}=\pi r^2$
$\displaystyle \frac{dA}{dt}=2\pi r \frac{dr}{dt}$
use the given to solve for the rate..

b)
$\displaystyle V_{slick}=\pi r^2 h$
this is just similar to the first item. the only difference is this is circular cylinder while the first one was a cone.. Ü