455
Answers
701–800
Answers and Explanations
726.
613 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 ].
The part of the chain x meters from the lifted end is raised 5 – x meters if 0 ≤ x ≤ 5 and
is lifted 0 meters otherwise. Therefore, the work required is
W
x
x
x dx
x x
n
i
i
n
=
−
(
)
=
−
=
−
=
→∞ =
∑
∫
lim
( )
(
)
*
5
4 9
49 5
49 5
2
4
1
0
5
2
0
5
∆
9 9 5 5
5
2
612 5
613
2
( )
( )
.
−
=
≈
J
J
727.
3,397,333 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 ].
Consider a horizontal slice of water that is Δ x meters thick and is at a height of x i
*
meters from the bottom of the tank. The trough has a triangular face with a width and
height of 4 meters, so by similar triangles, the width of the slice of water is the same as
the height x i
*
; therefore, the volume of the slice of water is 10 x
x
i
* ( )
( ) ∆ cubic meters.
To find the weight of the water, multiply the acceleration due to gravity (g = 9.8 m/s
2
)
by the mass, which equals the density of water (1,000 kg/m
3
) multiplied by the volume:
9 8
1 000
10
98 000
2
3
.
,
(
)
(
) ( ,
)
*
*
m/s
kg/m
m
m
m
(
)(
)
( )
(
) =
( )
x
x
x i
i
∆
(
( )
∆x N
The water must travel a distance of 7 −
(
)
x i
* meters to exit the tank. Therefore, the total
work required is
W
x
x
x
x
x dx
n
i
i
i
n
=
( ) −
(
)
=
−
=
→∞ =
∑
∫
lim ( ,
)
( )
,
( )(
)
*
*
98 000
7
98 000
7
1
0
4
∆
9 98 000 7
98 000 7
2
1
3
98 000 7
2
4 1
3
4
2
0
4
2
3
0
4
2
3
,
,
,
x x dx
x
x
−
(
)
=
−
(
)
=
−
(
)
=
∫
1 10 192 000
3
3 397 333
,
,
,
,
J
J
≈
Answers
701–800
Answers and Explanations
726.
613 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 ].
The part of the chain x meters from the lifted end is raised 5 – x meters if 0 ≤ x ≤ 5 and
is lifted 0 meters otherwise. Therefore, the work required is
W
x
x
x dx
x x
n
i
i
n
=
−
(
)
=
−
=
−
=
→∞ =
∑
∫
lim
( )
(
)
*
5
4 9
49 5
49 5
2
4
1
0
5
2
0
5
∆
9 9 5 5
5
2
612 5
613
2
( )
( )
.
−
=
≈
J
J
727.
3,397,333 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 ].
Consider a horizontal slice of water that is Δ x meters thick and is at a height of x i
*
meters from the bottom of the tank. The trough has a triangular face with a width and
height of 4 meters, so by similar triangles, the width of the slice of water is the same as
the height x i
*
; therefore, the volume of the slice of water is 10 x
x
i
* ( )
( ) ∆ cubic meters.
To find the weight of the water, multiply the acceleration due to gravity (g = 9.8 m/s
2
)
by the mass, which equals the density of water (1,000 kg/m
3
) multiplied by the volume:
9 8
1 000
10
98 000
2
3
.
,
(
)
(
) ( ,
)
*
*
m/s
kg/m
m
m
m
(
)(
)
( )
(
) =
( )
x
x
x i
i
∆
(
( )
∆x N
The water must travel a distance of 7 −
(
)
x i
* meters to exit the tank. Therefore, the total
work required is
W
x
x
x
x
x dx
n
i
i
i
n
=
( ) −
(
)
=
−
=
→∞ =
∑
∫
lim ( ,
)
( )
,
( )(
)
*
*
98 000
7
98 000
7
1
0
4
∆
9 98 000 7
98 000 7
2
1
3
98 000 7
2
4 1
3
4
2
0
4
2
3
0
4
2
3
,
,
,
x x dx
x
x
−
(
)
=
−
(
)
=
−
(
)
=
∫
1 10 192 000
3
3 397 333
,
,
,
,
J
J
≈