451
Answers
701–800
Answers and Explanations
718.
39,200 J
Let n be the number of subintervals of length Δ x, and let x i
* be a sample point in the ith
subinterval [x i–1 , x i ].
A horizontal slice of water that is Δ x meters thick and is at a distance of x i
* meters from
the top of the tank has a volume of ((4)(2)(Δ x)) cubic meters and has a mass of
mass density volume
kg/m
m
m
m
3
=
= (
)(
)
=
(
) (
)
,
(
)(
)(
)
,
1 000
4
2
8 0
∆x
0 00∆x kg
Because mass is not a force, you must multiply it by the acceleration due to gravity, 9.8
meters per second squared, in order to find the weight of the slice in newtons:
9 8
8 000
78 400
2
.
( ,
)
,
m/s
kg
N
(
)
=
∆
∆
x
x
The work required to pump out this slice of water is approximately 78 400
,
*
x
x
i
( ) ∆
joules. Therefore, the total work is
W
x
x
x dx
x
n
i
i
n
=
( )
=
=
=
→∞ =
∑
∫
lim
( ,
)
,
,
,
*
78 400
78 400
39 200
39 20
1
0
1
2
0
1
∆
0 0 J
719.
7.875 ft·lb
Begin by finding the spring constant, k, using the information about the work required
to stretch the spring 2 feet beyond its natural length:
W
kx dx
kx
k
k
k
=
=
=
−
(
)
=
=
∫ 0
2
2
0
2
2
2
14
2
14 2
2
0
14 2
7
( ) ( )
lb/ft
Next, find the work required to stretch the spring 18 inches, or 1.5 feet, beyond its
natural length:
W
xdx
x
=
=
=
−
(
)
=
⋅
∫ 7
7
2
7
2
1 5
0
7 875
0
1 5
2
0
1 5
2
2
.
.
( . ) ( )
.
ft lb