Work done pumping water up out of a trapezoid

The water trough is 5 feet long and its ends are trapezoids as shown. If the water trough is full of water, find work done in pumping the water out over the top. Assume that water weighs 62.5 lbs/ft^3.

In this problem, we're asked to calculate the work done in pumping water out of a trough that has trapezoidal ends. In physics, work is defined as the force applied over a distance. In the case of pumping water, the force is the weight of the water being moved, and the distance is how far the water needs to be lifted. This problem requires us to use the concept of work in a more complex, continuous setting, so we use calculus to break the problem down into smaller, manageable pieces.

To solve this problem, we first need to understand that the trough is five feet long and the ends are trapezoidal in shape. The key to solving this problem is to model the situation with calculus. We begin by writing an equation for the edge of the trough, which forms a straight line passing through two specific points: (1,0) and (1.5,2). This line describes how the width of the trough changes as you move along its length.

Next, we break the problem into slices. Imagine slicing the trough into thin horizontal slices of water, each slice with a height and width that corresponds to a small portion of the trough. For each slice, we calculate the volume and the weight, using the water’s weight density of 62.5 pounds per cubic foot. The volume of each slice is determined by the dimensions of the trapezoidal cross-section, which varies depending on the position along the length of the trough.

After determining the weight of the water in each slice, we calculate the work required to pump each slice out of the trough. This is done by multiplying the weight of the slice by the distance it needs to be lifted to reach the top of the trough. Since the trough is 5 feet long, the distance each slice must be lifted depends on its position within the trough. Finally, we sum the work for all the slices, which involves integrating over the length of the trough. The integral will give us the total work required to pump all of the water out of the trough.

This problem is a good example of how calculus can be applied to real-world physics problems, where we deal with continuous quantities like volume and force. Instead of calculating the total work all at once, we break it down into infinitesimally small slices, calculate the work for each slice, and then sum them up. This is the core idea of using integration to solve work problems in physics.

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