Question

A cylindrical tank has a radius of 4 m and a height of 10 m. The tank is filled with water. Find the work needed to pump the top 3 m of water out the top of the tank. (Use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)

Answer 1

Answer:

$w=2.216*10^{6} J$

Step-by-step explanation:

Data:

r=4m

h=10m

w=? (3 m of water out top of the tank)

g=9.8m/s2

б=1000kg/m3

We must find the work necessary to move 3m of water from the top of the tank, so there is a differential (dy) that represents a small portion of volume for which it is integrated for the total volume to move  then it would be the weight for the distance that is the volume to move

dw=weight*(10-y)dy, weight=б*V, , ⇒ weight=б*Vdy*g ⇒ dw=16πg(10-y)dy ⇒

$w=16\alpha \pi g(10y-\frac{y^{2}}{2})$ evaluated between a and b, but a=10 and b=7 (last 3m of the full tank), finally

$w=16\alpha \pi g((100-50)-(70-24.5))\\w=16\alpha \pi g(4.5)=72\alpha \pi g\\w=2.126*10^{6}J$