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Chapter 6: Problem 5

Why is integration used to find the work required to pump water out of a tank?

Short Answer

Expert verified
Answer: Integration is used to calculate the work required to pump water out of a tank because it allows us to sum up the work done for each infinitesimally small part of water at varying depths. This is necessary due to the force of gravity and the varying weight of water above each point in the tank. By integrating the work function with respect to the volume of water being pumped, we can calculate the total work required to empty the entire tank.

Step by step solution

01

Identify the relevant parameters in the problem

In this problem, we have to find the work required to pump water out of a tank. The key parameters we will need to examine are: the shape and dimensions of the tank, the weight of the water at each level, the depth of the water at each level, and the distance the water has to be moved vertically out of the tank.
02

Determine the weight of water at each level in the tank

Since the weight of water varies with depth, we first need to establish a relationship between the weight of water at each level in the tank. This can be expressed as a function w(x), where x is the depth of water in the tank. To do this, consider a thin horizontal slice of water at a depth x below the surface. The weight of water in this slice will be given by the product of the area of the slice, A(x), the thickness of the slice, dx, and the weight density of the water, denoted by 纬. Thus, we have: w(x) = A(x)纬dx
03

Express the work required to lift the water in terms of force and distance

The work, W, required to lift the water in the slice at a depth x to the top of the tank is given by the product of the force applied (which is equal to the weight of water) and the distance it has to be lifted (the depth x). We can express this as: dW = w(x) * x Combining this equation with the one from Step 2, we get: dW = A(x)纬xdx
04

Integrate the work required to lift the water

To find the total work necessary to pump all the water out of the tank, we must sum up the work done for each infinitesimal slice of water, meaning we need to integrate the work function with respect to the depth x from the bottom to the top of the tank. Let's denote the total height of the tank as H. The total work required to pump water out of the tank can be expressed as: W = 鈭個岽� (A(x)纬xdx) To solve this integration, we will need to determine the area function A(x) for the given tank shape, and then solve the integral. In summary, integration is used to find the work needed to pump water out of a tank because it allows us to sum up the work done for each infinitesimally small part of water at varying depths. The integration of the work function encompasses the shape and dimensions of the tank, the weight of water at each depth, and the distance the water must be lifted.

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