A right circular cylindrical water tank with height 30 meters and radius 5 meters is $\frac { 2 } { 3 }$ full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

A)98,000,000?
B)98,000?
C)9,800?
D)980?
E)98?

Question

A right circular cylindrical water tank with height 30 meters and radius 5 meters is

\frac { 2 } { 3 }

full. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in Newton-meters, is

A)98,000,000?
B)98,000?
C)9,800?
D)980?
E)98?

Quizplus · Accepted Answer

The work required to pump water is calculated by the formula $W = \rho g h A d$, where $\rho$ is the density of water (1000 kg/m$^3$), $g$ is the acceleration due to gravity (9.8 m/s$^2$), $h$ is the height to which the water is lifted, $A$ is the cross-sectional area of the water ($\pi r^2$, with $r = 5$ meters), and $d$ is the depth of water being pumped ($\frac{2}{3} \times 30 = 20$ meters). The work done to pump the water to the top of the tank varies with the depth, so we integrate from 0 to 20 meters, considering the changing height $h$ from the water surface to the top of the tank. The total work is given by integrating the force (weight of water) times the distance it needs to be lifted. Simplifying, we find the total work to be approximately 98,000,000 Newton-meters, considering the integral accounts for the varying depth of water and the constant need to lift it over the top of the tank.

A Right Circular Cylindrical Water Tank with Height 30 Meters 23\frac { 2 } { 3 }32​

A Right Circular Cylindrical Water Tank with Height 30 Meters $\frac { 2 } { 3 }$