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?
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:
So the ratio of radius to height is approximately .154