1. ## 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.

2. Originally Posted by shannon1111
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.
$\displaystyle \frac{r}{h} = \frac{2}{10}$

$\displaystyle W = \int walt$

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

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

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

$\displaystyle l = 10 - y$

$\displaystyle t = dy$

$\displaystyle W = \int_6^{10} 9800 \cdot \pi \left(\frac{h}{5}\right)^2(10-y) \, dy$

3. 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

4. Originally Posted by shannon1111
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