We can use the work-energy principle to find the work done in lifting the block of concrete. The work done by a constant force F over a distance d is given by:
W = Fd
In this case, the weight of the block of concrete is the force that must be overcome, and the distance lifted is either the distance x from the block to the top of the building or the distance y from the block to the ground. We know that the weight of the block is 550 lb, and the cable weighs 4 lb/ft, so the total weight being lifted is:
550 + 4x
or
550 + 4y
depending on which distance we choose. Using the work-energy principle, the work done in lifting the block is equal to the change in potential energy:
W = ∆PE = mgh
where m is the mass being lifted, g is the acceleration due to gravity, and h is the height through which the mass is lifted. In this case, the mass being lifted is the total weight being lifted, and h is either x or y, depending on which distance we choose. Substituting in the values we have, we get:
W = (550 + 4x)gh
or
W = (550 + 4y)gh
where g is the acceleration due to gravity, which is approximately 32.2 ft/s^2.
To express the work done as an integral in x, we need to integrate the expression for W with respect to x:
W = ∫(550 + 4x)gh dx
The limits of integration are from 0 to 400, since x represents the distance between the block and the top of the building. Therefore, the work done in lifting the block from the ground to the top of the building is:
W = ∫0^400 (550 + 4x)gh dx
To express the work done as an integral in y, we need to first express y in terms of x. We can use similar triangles to find the relationship between x and y. Let h be the height of the building, then:
y/h = x/(h + 400)
Solving for y, we get:
y = hx/(h + 400)
Substituting this expression for y into the expression for W, we get:
W = ∫0^x (550 + 4hx/(h + 400))gh dx
where the limits of integration are from 0 to 400, since when x = 0, y = 0, and when x = 400, y = 400h/(h + 400).