Question

an open tank has tge shape of circular cone. the tank is 8feet across the top and 6 feet high. watwr is pumped through the buttom of the tank. how much work is done to fill the tank from a depth of 4 feet to a depth of 6 feet? water weighs 62.4 pounds per cubic foot

Answer 1

`W=22652.98ft-lb`

Explanation:

We are given that

Height of tank=6feet

Diameter of tank=8 feet

Radius of cone,r=d/2=8/2=4 feet

Let a layer of thickness dy at distance yr from the bottom at origin

y=6 feet

Using similar triangle property

`(r)/(y)=(4)/(6)=(2)/(3)`

`r=(2)/(3)y`

Weight of water=62.4 pounds per cubic foot

Now,

Weight of layer=
`62.4* \pi r^2 h=62.4((4)/(9)\pi y^2 dy`

Now, work done to fill the tank from a depth of 4 feet to a depth of 6 feet is given by

`W=\int_(a)^(b)62.4* (4)/(9)\pi y^2(y) dy=\int_(4)^(6)62.4* (4)/(9)\pi y^3 dy`

`W=(62.4* 4\pi)/(9)[(y^4)/(4)]^(6)_(4)`

`W=(249.6\pi)/(36)(6^4-4^4)`

`W=22652.98ft-lb`