# Math Help - derivative word prob

1. ## derivative word prob

Water is leaking out of an inverted conical tank at a rate of 7200.000 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 8.000 meters and the diameter at the top is 5.000 meters. If the water level is rising at a rate of 29.000 centimeters per minute when the height of the water is 1.500 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Help is very much appreciated. Thanks.

2. ## Similar Triangles

Originally Posted by tennisgirl
Water is leaking out of an inverted conical tank at a rate of 7200.000 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 8.000 meters and the diameter at the top is 5.000 meters. If the water level is rising at a rate of 29.000 centimeters per minute when the height of the water is 1.500 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Help is very much appreciated. Thanks.

So the volume of the cone can be modeled by

$V=\frac{1}{3}\pi r^2h=\frac{1}{3}\pi \left( \frac{5h}{16}\right)^2h=\frac{25\pi}{768}h^3$

Now taking the derivative

$\frac{dV}{dt}=\frac{75\pi}{768}h^2\frac{dh}{dt}$

We know that $\frac{dV}{dt}=r_i-7200$ so we get...

$\frac{75\pi}{768}h^2\frac{dh}{dt}=r_i-7200 \iff r_i=\frac{75\pi}{768}h^2\frac{dh}{dt}+7200$

so finally we get...

$r_i=\frac{75\pi}{768}(150cm)^2(\frac{29cm}{s})+\fr ac{7200cm^3}{s}$

$r_i \approx \frac{200178cm^3}{s}+\frac{7200cm^3}{2}=\frac{2073 78cm^3}{s}$

3. ## Thanks!

Thank you so much. I wish you could tutor me. If you have AIM and are interested I'd hire you. Anyway, one more question:

what does "s" represent?

4. oh nevermind! The "s" is seconds! Thanks so much!