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

Answer:

$W=22652.98ft-lb$

Step-by-step 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

$\frac{r}{y}=\frac{4}{6}=\frac{2}{3}$

$r=\frac{2}{3}y$

Weight of water=62.4 pounds per cubic foot

Now,

Weight of layer=$62.4\times \pi r^2 h=62.4(\frac{4}{9}\pi y^2 dy$

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

$W=\int_{a}^{b}62.4\times \frac{4}{9}\pi y^2(y) dy=\int_{4}^{6}62.4\times \frac{4}{9}\pi y^3 dy$

$W=\frac{62.4\times 4\pi}{9}[\frac{y^4}{4}]^{6}_{4}$

$W=\frac{249.6\pi}{36}(6^4-4^4)$

$W=22652.98ft-lb$