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Chapter 7: Problem 19

Writing to Learn The cylindrical tank shown here is to be filled by pumping water from a lake 15 ft below the bottom of the tank. There are two ways to go about this. One is to pump the water through a hose attached to a valve in the bottom of the tank. The other is to attach the hose to the rim of the tank and let the water pour in. Which way will require less work? Give reasons for your answer.

Short Answer

Expert verified
The first method, where water is pumped through a hose attached to the bottom of the tank, will require less work as the distance that the water needs to be moved is lesser compared to the second method.

Step by step solution

01

- Understand the physical principles involved

The fundamental physics principle to understand is that the work done (W) to move an object (in this case, water), is the product of the force (F) used and the distance (d) over which the force is applied. In mathematical terms, we say \( W = F \cdot d \). The force to move the water is the same in both cases because regardless of how it is moved, the water weighs the same. The only variable is the distance the water needs to traverse.
02

- Analyze the first method

In the first method, the water has to be pumped through a hose attached to the bottom of the tank. So the distance it needs to be moved is the height of the tank (h) plus the 15 feet from the lake to the bottom of the tank. Thus, the total distance is \( h + 15 \) feet.
03

- Analyze the second method

In the second method, the water is pumped to the rim of the tank, which means the distance it has to be moved is higher by the height of the tank. So, the total distance in this case is \( 2h + 15 \) feet.
04

- Compare the work to be done in both methods

Comparing the work done in both methods, it becomes evident that the first method would require less work as the distance the water needs to traverse is less. This can be mathematically expressed as \( W_1 = F \cdot (h + 15) \) for the first method versus \( W_2 = F \cdot (2h + 15) \) for the second method. Given that \( h > 0 \), it is immediately clear that \( W_1 < W_2 \).

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