A cable weighing 2 lbs/ft is used to lift a 300 lbs load from the bottom of a well 100 feet deep.
Calculate the work done to raise the load to the surface.

The string:
(a) S₇P₃A₃₁ (Force - Pull).
The math:

Pj Problem of interest is of type force. Work problems are generally of type force because force is the doer of work. In the case of white-collar work, intellectual force actuates materiality.
Consider the above diagram (fig. CW.1):
Let W be the work done to raise the load x feet.
Let k be the work done in raising the load h feet.
k = [300 + 2(100 - x_avg)]h -------(1). x_avg is the representative average of the partitions of the distance BC.
So, the rate of change of k with respect to h is:
k/h = [300 + 2(100 - x_avg)] = 500 -2x_avg
limit (k/h) as h → 0 = dW/dx (instantaneous change of W with respect to x)
So, limit (k/h) as h → 0 = 500 -2x. Since x_avg → x.
So, dW/dx = 500 -2x.
So, W = 500x - x² + C---------(2). Where C is a constant.
When x = 0, W = 0. So C = 0.
So, W = 500x - x².
So, for x = 100,
W = 500(100) - (100)(100) = 40,000 ft-lb.

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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