Solve the problem.
-A cylinder with height 9 m and radius 3 m is filled with water and must be emptied through an outlet pipe 3 m above the top of the cylinder. Compute the work required to empty the water in the top half of the tank. Then compute the work required to empty the (equal amount of) water in the lower half of the tank. Round to two decimal places when appropriate.
A) 11,222,083.1 J; 5,611,041.55 J
B) 5,611,041.55 J; 5,611,041.55 J
C) 6,546,215.14 J; 12,157,256.7 J
D) 18,703,471.8 J; 12,157,256.7 J

Question

Solve the problem.   
-A cylinder with height 9 m and radius 3 m is filled with water and must be emptied through an outlet pipe 3 m above the top of the cylinder. Compute the work required to empty the water in the top half of the tank. Then compute the work required to empty the (equal amount of) water in the lower half of the tank. Round to two decimal places when appropriate.
A) 11,222,083.1 J; 5,611,041.55 J
B) 5,611,041.55 J; 5,611,041.55 J
C) 6,546,215.14 J; 12,157,256.7 J
D) 18,703,471.8 J; 12,157,256.7 J

Quizplus · Accepted Answer

To solve this problem, we use the concept of work done in moving a volume of water against gravity, which is given by the formula $W = \rho g h \Delta V$, where $\rho$ is the density of water (approximately $1000 \, \text{kg/m}^3$), $g$ is the acceleration due to gravity ($9.8 \, \text{m/s}^2$), $h$ is the height through which the water is lifted, and $\Delta V$ is the volume of water moved.For the top half of the tank, the water needs to be lifted a shorter distance on average compared to the bottom half. The volume of water in each half is the same since the tank is a cylinder and we're dividing it into equal halves by height. The volume of the top half is $\pi r^2 \frac{h}{2}$, where $r = 3 \, \text{m}$ and $h = 9 \, \text{m}$, so each half has a volume of $\pi \cdot 3^2 \cdot \frac{9}{2} = 42.41 \, \text{m}^3$.For the top half, the average height the water needs to be lifted is less since it starts closer to the outlet, while for the bottom half, the water needs to be lifted a greater average height. Specifically, the water from the bottom half needs to be lifted an additional 4.5 m on average (the midpoint of the bottom half) compared to the top half.Given these considerations, the work required for each half will differ due to the difference in lifting heights. The exact calculations involve integrating the work done to lift each infinitesimal volume of water from its initial position to the outlet, taking into account the changing height as you move from the bottom to the top of the tank. The result shows that it requires more work to empty the bottom half of the tank compared to the top half, due to the greater average height through which the water must be lifted. This aligns with choice C, which indicates different work values for the top and bottom halves, reflecting the additional effort needed to lift water from the lower half of the tank to the outlet.

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