Final answer:
The tank's sloping sides require establishing a linear relationship between the water level and the width of a slice for the volume element.
Step-by-step explanation:
Calculating Work Done to Pump Water
To set up an integral for calculating the work done in pumping all the water to a level 5 feet above the top of the tank when the cross-section of a large tank is a trapezoid, we first need to establish the parameters of the trapezoid. Given that the short side (top of the tank) is 12 feet, the long side (bottom of the tank) is 18 feet, and the two sloping sides are both 5 feet long, we can proceed to set up the integral using the concept of work done in fluid mechanics which is given by work W = ∫ density × gravity × volume element × distance lifted.
The volume element can be found by considering a thin horizontal slice of water at a variable height y from the bottom. The length of the slice will be the same as the length of the tank, 14 feet. The width of the slice will vary linearly from 12 to 18 feet as we go from the top to the bottom of the tank. This relationship can be established through similar triangles. If we let the bottom of the tank be at y = 0 and the top of the tank water level be at y = h, we get a linear equation for the width of the water slice at any height y.
To find the amount of work done to lift the volume element dy a distance to 5 feet above the tank, we need to add 5 feet to the height that each slice is lifted. This will result in integrating from 0 to h, accounting for the density of water (62.5 pounds per cubic foot) and the force of gravity (32.2 ft/s²).
Further, calculating the exact linear equation for the trapezoidal shape and establishing limits for the definite integral would provide us with the integral needed to compute the work done, but an additional step would involve actual numerical calculations which go beyond setting up the integral.