# Optimization question - water in a conical tank

1. Oct 11, 2008

### billmccai

1. The problem statement, all variables and given/known data

A water tank is in the shape of an inverted conical cone with top radius of 20m and

depth of 15m. Water is flowing into the tank at a rate of 0.1m^3/min.

(a) How fast is the depth of water in the tank increasing when the depth is 5m?

Water is now leaking from the tank at a rate that depends on the depth h, (h= height of

water in the tank) this rate is 0.1h^3/min.

(b) How fast is the depth of water in the tank changing when the depth is 5m?

(c) How full can the tank get?

2. Relevant equations

3. The attempt at a solution

Ok for part A:

Tan(angle) = 20/15 = 3/4

So i got a formula for r.... r = 4/3 h

Which i put into the formula for the volume of a cone and got:

V = 16pi/27 * h^3

and then differentiated V with respect to time.

dV/dt = 16pi/27 * (3h^2)dh/dt

and i know h and dV/dt so i subbed those in and got

dh/dt = 7.1697 * 10^(-4) m/min

which i'm pretty sure is right.

Now I'm not sure how to do part b.

Do I set dV/dt as 0.1 - 0.1h^3

and then just do the same thing? Or is this wrong. Also, what about part c?