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Math Help - Work: Inverted Cone

  1. #1
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    Work: Inverted Cone

    A tank has the shape of an inverted circular cone with height H=10m and base radius R=2m. The tank is initially full of water.

    Find the work required to lower the height of the water level to h=6m by pumping the water to the top of the tank.

    The density of water is r= 1000 kg/m3 and the acceleration due to gravity is g = 9.8 m/s2.
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    Quote Originally Posted by shannon1111 View Post
    A tank has the shape of an inverted circular cone with height H=10m and base radius R=2m. The tank is initially full of water.

    Find the work required to lower the height of the water level to h=6m by pumping the water to the top of the tank.

    The density of water is r= 1000 kg/m3 and the acceleration due to gravity is g = 9.8 m/s2.
    \frac{r}{h} = \frac{2}{10}

    W = \int walt

    w = weight density
    a = cross-sectional area of a representative horizontal slice of liquid
    l = "lift" distance of a representative horizontal slice of liquid
    t = slice thickness


    w = \rho g = 1000 \cdot 9.8 = 9800 \, N/m^3

    a = \pi \cdot r^2 = \pi \left(\frac{h}{5}\right)^2

    l = 10 - y

    t = dy

    W = \int_6^{10} 9800 \cdot \pi \left(\frac{h}{5}\right)^2(10-y) \, dy
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    To find the work required to lower the height of the water level to h=6m we need to integrate the function W(x) that, for each 0 x H-h, represents the work needed to raise the layer that is x meters below the top to the top of the tank. so what is the shape of such layer? cone? I am not sure for that
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    Quote Originally Posted by shannon1111 View Post
    To find the work required to lower the height of the water level to h=6m we need to integrate the function W(x) that, for each 0 x H-h, represents the work needed to raise the layer that is x meters below the top to the top of the tank. so what is the shape of such layer? cone? I am not sure for that

    oh,sorry ,I just got it , it's a circle
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