Chapter 11.2, Problem 50E

Solution

To find: The amount of the work done in lifting the barrel to a height of 35 ft.

The total work done to lift the barrel to 35 ft is 33250 ft-lb.

Given information:

The weight of the water barrel is 1250 lb. It has a significant leak and must be lifted 35ft.

The weight of the water barrel for different values in the interval $\left[0,30\right]$ are given.

Formula used:

The area of the rectangle is $\text{base}\times \text{height}$ .

Left Rectangular Approximation Method (LRAM):

Divide an interval $\left[a,b\right]$ into $n$ subintervals of length $\Delta x=\frac{\left(b-a\right)}{n}$ . The subintervals are $\left[a,a+\Delta x\right]$ , $\left[a+\Delta x,a+2\Delta x\right],$ …, $\left[a+\left(n-1\right)\Delta x,a+n\Delta x=b\right]$ . The area under the curve approximated as the sum of the products defined below:

  ${\displaystyle \sum _{i=0}^{n-1}f\left(a+i\Delta x\right)\Delta x}$

Calculation:

Divide the interval $\left[0,30\right]$

into seven sub-intervals. The velocity of the right endpoints of the subintervals is considered as the height of the rectangle and the base of each rectangle is 5 ft. The area of seven rectangles and the total area are computed in the table below:

Distance (ft)	Weight (lb)	Work
$\left[0,5\right]$	1250	6250
$\left[5,10\right]$	1150	5750
$\left[10,15\right]$	1050	5250
$\left[15,20\right]$	950	4750
$\left[20,25\right]$	850	4250
$\left[25,30\right]$	750	3750
$\left[30,35\right]$	650	3250
	Total Work	33250

Thus, the total work done to lift the barrel to 35 ft is 33250 ft-lb.