Class Activity #3
David Amos
MATH 152 – Engineering Math II, Spring 2015
1
1.1
Work
Example 1
A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of the building 120 ft
high. How much work is done in pulling the rope to the top of the building?
Solution. We divide the rope into peices of length ∆x. The portion of the rope from x ft
to x + ∆x feet below the top of the building weighs 0.5 ft/lb and must be moved x feet.
The direction of the force (the weight) is in the same direction as the rope is hanging, and
so from the formula work = force · distance, we get that the amount of work contributed
by this piece is 12 x∆x. Thus the total work is given by
Z 50
1 2 50
1
xdx = x = 625 ft-lb.
2
4 0
0
1.2
Example 2
A uniform cable hanging over the edge of a tall building is 40 ft long and weighs 60 lbs.
How much work is required to pull 10 ft of the cable to the top?
Solution. Each part of the 10 ft of rope that get pulled up and over the edge is moved
an equal distance to its distance from the top. The remaining 30 ft move 10 ft up. Thus,
since the rope way 60/40 = 3/2 lb/ft, the total work done is
Z 10
Z 40
40
3
3
3 10
xdx +
· 10dx = x2 + 15x = 525 ft-lb.
2
4 0
10
0
10 2
1.3
Problems
Problem 1. A cable that weighs 2 ft/lb is used to lift 800 lbs of coal up a mineshaft 500
ft deep. Find the work done.
1
D. Amos
1.4
Class Activity #3
2
Example 3
An aquarium 2m long, 1m wide, and 1m deep is full of water. Find the work needed to
pump half of the water out of the aquarium.
Solution. A “slice” of water ∆x thick lying at a depth of x meters (where 0 ≤ x ≤ 21 has
volume 2∆x m3 . Since mass = density × volume, and the density of water is 1000 kg/m3 ,
the slice has a mass of 2000∆x kg, and therefore weighs about (9.8)(2000∆x = 19, 600∆x
N. This weight must be moved through a distance of x meters, so the work contributed
by the slice is 19, 600x∆x J. Thus the total work is given by
Z
0
1.5
1/2
1/2
19, 600xdx = 9, 800x = 2450 J.
2
0
Example 4
A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth
of water is 4ft. How much work is required to pump all of the water out over the side?
Solution. A horizontal cylindrical slice of water ∆x thick has a volume of πr2 h = π · 122 ·
∆x ft3 , and weighs about (62.5 lb/ft3 )(144π∆x gt3 ) = 9000π∆x lb. The work needed to
pump the slice out, assuming the slice lies x ft below the edge of the pool, is 9000πx∆x
ft-lb. Thus the total work done is
Z 4
4
9000πxdx = 4500πx2 = 108, 000π ft-lb.
1
1.6
1
Problems
Suppose that the tank shown below is full of water. Find the work required to pump the
water out of the outlet (recall that the density of water is 1000 kg). Use a = 3 m, b = 3
m, c = 8 m, d = 2 m.
D. Amos
2
2.1
Class Activity #3
3
The average value of a function
Example 1
Find the average value of f (x) = 1/x on the interval [1, 4].
Solution. The average value is given by
Z 4
4 ln(4)
1
1
1
dx = ln(x) =
.
4−1 1 x
3
3
1
2.2
Example 2
Find c such that favg = f (c), where f (x) = 4x − x2 , on the interval [0, 3].
Solution. The average value is given by
Z 3
1 2 1 3 3 1
1
(4x − x2 )dx =
2x − x = (18 − 9) = 3.
3−0 0
3
3
3
0
To find c such that f (c) = 3, set 4c − c2 = 3 and solve for c. This gives 0 = c2 − 4c + 3 =
(c − 1)(c − 3), so that the values for c are 1 and 3 (note we do not throw out any values
since both occur in out interval).
2.3
Problems
Problem 2. The temperature of a metal rod, 5 m long, is 4x (in ◦ C) at a distance x
meters from one end of the rod. What is the average temperature of the rod?
Problem 3. Find the numbers b such that the average value of f (x) = 2 + 6x − 3x2 on
the interval [0, b] is equal to 3.