AP CALCULUS BC NOTES
WORK PROBLEMS
Sample Problem:
1. A cylindrical reservoir of diameter 4 ft and height 6 ft is half–full of water weighing 10 lb/ft3.
Find the work done in emptying the water over the top.
Answer:
The volume of a slice of water is ∆V = πx2 ∆y, where x = 2.
A slice at height y is lifted (6 – y) ft.
3
So ∆W = 10 • π • 4 ∆y (6 – y); So W = 10•4π
∫ ( 6 − y ) dy = 540π
ft–lb.
0
(We used 3 as the upper limit since the reservoir is only half full.)
OR
Work = Force (weight) X Distance
ft–lb =
Water weighs 10 lb/ft 3
lb
• ft 3 • ft = 10 • π (2) 2 h • (6 − y )
ft 3
3
So, W = 40π ∫ (6 − y )dy = 540π ft–lb
0
(See picture below.)
BROSE
3/17/2011
AP CALCULUS BC NOTES
WORK PROBLEMS
1. A rectangular tank with a base 4 feet by 5 feet
and a height of 4 feet is full of water (see picture).
The water weighs 62.4 pounds per cubic foot).
PROBLEM #1
a) How much work is done in pumping water out
over the top edge in order to empty ALL of the
tank.
b) How much work is done in pumping water out
over the top edge in order to empty HALF of the
tank.
2. A cylindrical water tank 4 meters high with a
radius of 2 meters is buried so that the top of the
tank is 1 meter below ground level (see picture).
The weight of water is 1000 kilograms per cubic
meter. How much work is done in pumping a full
tank of water up to ground level?
3. The vat shown below contains water to a depth
of 2 meters. Find the work required to pump all
the water to the top of the vat.
PROBLEM #2
PROBLEM #3
[Use 9810 N/m3 as the weight density of water].
Hint: you will have to use similar triangles to
solve this problem.
ANSWERS: 1a) 9,984 foot-pounds 1b) 2,496 foot-pounds 2) 48,000π kg-meters 3) 261,600 joules
BROSE
3/17/2011