A coffee filter has the shape of an inverted cone. Water drains out of the filter at a rate of 8cm^3/s. When the depth of water in the cone is 6cm..., the depth is decreasing at a rate of 3cm/s....

A coffee filter has the shape of an inverted cone. Water drains out of the filter at a rate of 8cm^3/s. When the depth of water in the cone is 6cm...

, the depth is decreasing at a rate of 3cm/s. What is the ratio of the radius of the cone to its height?

Asked on by islnds

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Let R be the radius of the cone, and H the height of the cone. Then let r be the radius at the depth of the water, and h the depth of the water.

By triangle similarity we have `R/H=r/h` . Let `R/H=k` so that `r=kh` .

The volume V of the cone is given by `V=1/3 pi r^2h` . Substituting for r we get `V=1/3 pi (kh)^2h=1/3 pi k^2h^3` where k is a constant.

We are given that `(dV)/(dt)=-8` , h=6, and `(dh)/(dt)=-3` .

`(dV)/(dt)=k^2pih^2(dh)/(dt)` ;substituting known values we get:

`-8=k^2pi(36)(-3)=> k=sqrt(2/(27pi))~~.154`

So the ratio of radius to height is approximately .154

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