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Optimization question - water in a conical tank

  1. Oct 11, 2008 #1
    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?

    I tend to get a bit lost with these sorts of wordy questions. Please help.
  2. jcsd
  3. Oct 11, 2008 #2


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    Staff Emeritus
    Science Advisor
    Homework Helper

    (a) Correct
    (b) Yes, that's the right idea.
    (c) If you're still stuck after completing (b), post again.
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