Related Rates Problem

**EliteNewbz** · January 19th 2010, 07:42 PM

I really don't understand related rates so I'm having a problem with this question.

The top of a silo has the shape of a hemisphere of diameter 20 feet. If it is coated uniformly with a layer of ice and if the thickness is decreasing at a rate of 1/4 inch/hr, how fast is the volume of the ice changing when the ice is 2 inches thick?

So I basically got to

dr = 1/4 (.25)
dt

When we're looking for dv/dt ? (Volume?)

So I assumed -> 4(pi)r^2(dr/dt) = dv/dt

and I'm pretty lost... :/

**drumist** · January 19th 2010, 08:04 PM

The hard part is setting up the formula for the volume of the ice. We want the volume of the ice only, not the ice plus the silo. This ends up being the difference between two hemispheres -- the larger one (silo + ice) minus the smaller one (just the silo), thus leaving us with the ice only. So the equation is:

$<br /> V=\frac{2 \pi}{3} (10+x)^3 - \frac{2 \pi}{3} 10^3<br />$

where $x$ is the thickness of the ice. Also, make sure we define $\frac{dx}{dt}=-1/4$ because it is decreasing.

At this point you just need to differentiate and start plugging in the appropriate values and solve.

**EliteNewbz** · January 19th 2010, 08:10 PM

Ohhhhhhhhh, Thanks I get it. I was using the wrong formula as well -_-. HEMI-Sphere.

**drumist** · January 19th 2010, 08:25 PM

Oops, I made a mistake earlier. It should be raised to the 3rd power, not 2nd power. I fixed it above.

Good luck.

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