A bucket is being lifted from the bottom of a 50-foot deep well; its weight (including the water), $B$, in pounds at a height $h$ feet above the water is given by the function $B(h)$. 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 $B(h) =$

(b) Compute the value of the product $B(5) \triangle h$, where $\triangle h = 2$ feet. Notice that this product represents the approximate work it took to move the bucket of water from $h = 5$ to $h = 7$.

$B(5) \triangle h =$

What are the units on the product $B(5) \triangle h =$?

(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 $h = 7$? Think about why your answer is true.

(d) Compute the value of the product $B(22) \triangle h$, where $\triangle h = 0.25$ feet.

$B(22) \triangle h =$

What are the units on the product $B(22) \triangle h =$?

(e) Notice that the value in part (d) estimates the amount of work it takes to move the bucket from a height of 22 feet to 22.25 feet. More generally, the quantity $W_{\small\mbox{slice}} = B(h) \triangle h$ measures the amount of work it takes to move from a given value of $h$ to a small positive value $\triangle h$.

(f) Evaluate the definite integral $\int_0^{50} B(h) \, dh =$ .

What is the meaning of the value you find? Why?

You can earn partial credit on this problem.