# related rates

• Jul 20th 2008, 08:41 PM
zintore
related rates - still need help!
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 0.02 meter. At that moment:

(a) 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?

the answer to (a) is (i hope)
dA/dt= 2(pi)r (dA/dt)
dA/dt|r=150 = 30pi

so for (b).. the area of a cylinder is the area of the top times the height, but i dont know to go from there... do i find dh/dt or somethin? how do i do that?
• Jul 20th 2008, 09:00 PM
kalagota
Quote:

Originally Posted by zintore
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 0.02 meter. At that moment:

(a) 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?

the answer to (a) is (i hope)
dA/dt= 2(pi)r (dA/dt) should have been (dr/dt) dA/dt|r=150 = 30pi correct!

so for (b).. the area of a cylinder is the area of the top times the height, but i dont know to go from there... do i find dh/dt or somethin? how do i do that?

note that: $\displaystyle V_{cyl}=\pi r^2 h$ and $\displaystyle V$ is contant

so, $\displaystyle \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} + 2\pi r h \frac{dr}{dt}$

can you do from here?
• Jul 20th 2008, 09:12 PM
zintore
umm
sorry i still dont understand, how do i find dr/dt?
• Jul 20th 2008, 09:22 PM
kalagota
it was already given:
Quote:

the radius of the slick is expanding by 0.1 meter/minute
• Jul 20th 2008, 09:37 PM
zintore
hmm
ok now i have dV/dt= 3pi/5 + 450pi dh/dt
do i answer it in terms of dV/dt like.. (dV/dt -3pi/5)/450pi or... is there a way to solve dV/dt?
• Jul 20th 2008, 10:26 PM
kalagota
it was also given..

note that
Quote:

the volume of oil spilled remains fixed
that is, the volume is constant.. and what is the derivative of a constant? :)