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Math Help - [SOLVED] Find rate of increase of water level?

  1. #1
    Super Member fardeen_gen's Avatar
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    [SOLVED] Find rate of increase of water level?

    Water runs into a conical tank at the rate of 9 ft^3/ min. The tank stands point down and has height of 10 ft and a base radius of 5 ft. How fast in the water level rising when water is 6 feet deep?

    What is the answer? Please explain the steps also.
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  2. #2
    Eater of Worlds
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    This is an old cliche related rates problem. Do a site search and many will pop up.
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  3. #3
    Lord of certain Rings
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    Quote Originally Posted by fardeen_gen View Post
    Water runs into a conical tank at the rate of 9 ft^3/ min. The tank stands point down and has height of 10 ft and a base radius of 5 ft. How fast in the water level rising when water is 6 feet deep?

    What is the answer? Please explain the steps also.
    V = \frac13 \pi r^2 h

    Now when the water is at a level of h, let the surface water radius be r. Now if you draw a figure, you will see that if the half angle of the cone is \theta, then \tan \theta = \frac{5}{10} = \frac12 = \frac{r}{h}

    Thus V = \frac13 \pi \left(\frac{h}2\right)^2 h = \frac{\pi h^3}{12} \Rightarrow \frac{dV}{dt} = \frac{3\pi h^2}{12}\bigg{|}_{h=6}\frac{dh}{dt}

    9 = \frac{\pi 6^2}{4}\frac{dh}{dt} \Rightarrow \frac{dh}{dt} = \frac1{\pi} \text{ ft/min}
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