# Math Help - Rate Question

1. ## Rate Question

A liquid is going through a conical filter whose top diameter is the same as its height. It pours into a right circular cylinder whose radius is 20cm. How fast is the liquid in the cylinder rising in cm/min if the liquid in the filter is 10 cm high and falling at a rate of 80cm/min?

I am not even sure where to start this question, any tips/help would be appreciated.

2. One thing nice about cylinders is that the rate of change of height remains the same because of its straight sides.

Cone volume = $\frac{\pi}{3}r^{2}h$

But we are told 2r=h, so sub:

$V=\frac{\pi}{3}(\frac{h}{2})^{2}h$

$V=\frac{{\pi}h^{3}}{12}$

Differentiate w.r.t h:

$\frac{dV}{dt}=\frac{{\pi}h^{2}}{4}\cdot\frac{dh}{d t}$

We are told dh/dt=-80 and h=10.

$\frac{{\pi}(10)^{2}}{4}\cdot (-80)=-2000{\pi}$

Therefore, the volume of the cylinder is filling up at a rate of $2000{\pi}$ cubic cm per min.

Since the cylinder has a 20 cm radius, it's volume is $V={\pi}r^{2}h$

$2000{\pi}=400{\pi}h$

$h=5 \;\ cm/min$

3. Originally Posted by CalcGeek31
A liquid is going through a conical filter whose top diameter is the same as its height. It pours into a right circular cylinder whose radius is 20cm. How fast is the liquid in the cylinder rising in cm/min if the liquid in the filter is 10 cm high and falling at a rate of 80cm/min?

I am not even sure where to start this question, any tips/help would be appreciated.
A good place to start might be with the equations.

What is the formula for the volume V of a cone with height "H" and radius "R"? Noting that, in this case, H = R what is the formula that you should use?

What is the formula for the volume v of a cylinder with radius "r" and height "h"? Noting that, in this case, r = 20, what is the formula that you should use?

You are given that, for the cone, dH/dt = -80, and are asked to find the rate of change dh/dt when H = 10. Can you think of a way to relate the various bits of information?