related rates

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July 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?
July 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: $V_{cyl}=\pi r^2 h$ and $V$ is contant

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

can you do from here?
July 20th 2008, 09:12 PM
zintore

umm

sorry i still dont understand, how do i find dr/dt?
July 20th 2008, 09:22 PM
kalagota

it was already given:

Quote:

the radius of the slick is expanding by 0.1 meter/minute
July 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?
July 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? :)