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Math Help - Related Rates Problem

  1. #1
    Newbie
    Joined
    Nov 2007
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    4

    Related Rates Problem

    Hi,I'm having some trouble with a related rate problem. A pool is 40ft long, 20 ft wide, 4 ft deep at the shallow end and 9 ft deep at the deep end. Water is being pumped into the pool at 10 ft^3/min. How fast is the water level rising when there is 4 ft of water in the deep end?

    I've tried using volume of a trapezoidal prism and differentiating that, but it didn't seem to work. Any help is appreciated. Thanks a lot.
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  2. #2
    MHF Contributor
    Joined
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    Since the question is for dh/dt when h=4ft at the 9-ft deep end, then the volume of a trapezoidal prism won't work. Because at h=4ft, you only have a triangular prism for the volume of water in the tank.

    So,
    Draw the figure, a longitudinal section of the pool, such that the walls are vertical. It is a trapezoid, (trapezium).
    Draw horizontal line passing through the 4-ft deep.
    Draw another horizontal line lower than tha first line.
    Two similar right triangles are formed.

    One has:
    ----vertical leg = (9-4) = 5ft
    ----horizontal leg = 40ft
    ----hypotenuse---unknown.

    The smaller one has:
    ---vertical leg = h
    ---horizontal leg = x
    ---hypotenuse----unknown too.

    By proportion,
    5/40 = h/x
    Cross multiply,
    5x = 40h
    x = 40h/5 = 8h

    So, volume of water in the pool when depth at deep end is h ft,
    V = (1/2)(h)(x) *(20)
    V = 10h(8h)
    V = 80h^2

    dV/dt = 80(2h *dh/dt)
    dV/dt = 160h (dh/dt)

    When h=4, and dV/dt is always 10 cuft/min,
    10 = 160(4)(dh/dt)
    1 = 64(dh/dt)
    dh/dt = 1/64

    Therefore, at that instant, the water is rising at the rate of 1/64 ft/min. ----answer.
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  3. #3
    Super Member

    Joined
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    Lexington, MA (USA)
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    Hello, HclBr!

    ticbol did an excellent job of explaining the solution.

    Here are some diagrams . . .


    This is the entire pool.
    Code:
                             40
                *-----------------------------*
           20 /                             / |
            /                             /   |4
          *-----------------------------*     |
          |                             |     *
         4|                            4|   /
          |                             | /
          + - - - - - - - - - - - - - - *
          |                       *
         5|                 *
          |           *
          |     *
          *


    This is the lower portion with the water.
    Code:
                              40 
          *-----------------------------------------------*
          |                                         *
          |            x                      *
          | - - - - - - - - - - - - - - *
        5 | : : : : : : : : : : : *
          |h : : : : : : : :*
          | : : : : : *
          |: : :*
          *
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