Question

A conical tank is full of water is 16 feet across the top and 12 feet deep. Find the work required to pump all the water to a point 2 feet above the top of the vessel.

Answer 1

work done = force $\times distace$

= weight density $\times distance = mgd = fd$

= $\rho Vgd$

= $\int_{a}^{b} \Delta x dx$

taking a small porting of tank to be cylindrical

v= $\pi r^2h$

$\frac{R}{16}= \frac{12-x}{12}$

R= $\frac{12-x}{12} \times 16 = \frac{48-4x}{3}$

$V= (\frac {48-4x}{13})^2 \times \pi dx$

= $( 256- \frac{128x}{3}+\frac{32x^2}{9})\times \pi$

$\Delta = x$

V= $804.24x^2-134.04x^3+11.17x^4$

work done w= $\rho \int_{2}^{12} 804.24x^2-134.04x^3+11.17x^4$

w= $1.795\times 10^7 Nfeet$