Active Calculus 2nd Ed

Suppose that a heavy rope hangs over the side of a cliff. The rope is 200 feet long and weighs 0.3 pounds per foot; initially the rope is fully extended. How much work is required to haul in the entire length of the rope? (Hint: set up a function \(F(h)\) whose value is the weight of the rope remaining over the cliff after \(h\) feet have been hauled in.)

A leaky bucket is being hauled up from a 100 foot deep well. When lifted from the water, the bucket and water together weigh 40 pounds. As the bucket is being hauled upward at a constant rate, the bucket leaks water at a constant rate so that it is losing weight at a rate of 0.1 pounds per foot. What function \(B(h)\) tells the weight of the bucket after the bucket has been lifted \(h\) feet? What is the total amount of work accomplished in lifting the bucket to the top of the well?

Now suppose that the bucket in (b) does not leak at a constant rate, but rather that its weight at a height \(h\) feet above the water is given by \(B(h) = 25 + 15e^{-0.05h}\text{.}\) What is the total work required to lift the bucket 100 feet? What is the average force exerted on the bucket on the interval \(h = 0\) to \(h = 100\text{?}\)

From physics, Hooke’s Law for springs states that the amount of force required to hold a spring that is compressed (or extended) to a particular length is proportionate to the distance the spring is compressed (or extended) from its natural length. That is, the force to compress (or extend) a spring \(x\) units from its natural length is \(F(x) = kx\) for some constant \(k\) (which is called the spring constant.) For springs, we choose to measure the force in pounds and the distance the spring is compressed in feet. Suppose that a force of 5 pounds extends a particular spring 4 inches (1/3 foot) beyond its natural length.