Preview Activity 6.4.1.
A bucket is being lifted from the bottom of a 50-foot deep well; its weight (including the water), \(B\text{,}\) in pounds at a height \(h\) feet above the water is given by the function \(B(h)\text{.}\) When the bucket leaves the water, the bucket and water together weigh \(B(0) = 20\) pounds, and when the bucket reaches the top of the well, \(B(50) = 12\) pounds. Assume that the bucket loses water at a constant rate (as a function of height, \(h\)) throughout its journey from the bottom to the top of the well.
(a)
Find a formula for \(B(h)\text{.}\)
(b)
Compute the value of the product \(B(5) \Delta h\text{,}\) where \(\Delta h = 2\) feet. Include units on your answer. Explain why this product represents the approximate work it took to move the bucket of water from \(h = 5\) to \(h = 7\text{.}\)
(c)
Is the value in (b) an over- or under-estimate of the actual amount of work it took to move the bucket from \(h = 5\) to \(h = 7\text{?}\) Why?
(d)
Compute the value of the product \(B(22) \Delta h\text{,}\) where \(\Delta h = 0.25\) feet. Include units on your answer. What is the meaning of the value you found?
(e)
More generally, what does the quantity \(W_{\text{slice} } = B(h) \Delta h\) measure for a given value of \(h\) and a small positive value of \(\Delta h\text{?}\)
(f)
Evaluate the definite integral \(\int_0^{50} B(h) \, dh\text{.}\) What is the meaning of the value you find? Why?
