461
Answers
701–800
Answers and Explanations
Let n be the number of subintervals of length Δ x and let x i
* be a sample point in the ith
subinterval. Consider a horizontal slice of water that is Δ x feet thick and is at a
distance of x i
* feet from the top of the tank; let w i
* be the radius of the slice of water.
Because the tank has a shape of a hemisphere of radius 4, you have the following
relationship:
x
w
i
i
*
*
( ) + ( ) =
2
2
16
Therefore, the slice has a volume of
V
w
x
x
x
i
i
i
= ( ) =
− ( )
(
)
π
π
*
*
2
2
3
16
∆
∆ ft
And the force on this slice is
F
x
x
i
i =
− ( )
(
)
( . )
*
62 5
16
2
π
∆ lb
The slice of water must travel a distance of 1 +
(
)
x i
* feet to exit the tank. Therefore, the
total work required is
W
x
x
x
x
x
n
i
i
i
n
=
+
(
) − ( )
(
)
=
+
−
(
)
→∞ =
∑
lim ( . )
.
(
)
*
*
62 5
1
16
62 5
1
16
2
1
2
π
π
∆
d dx
x x
x dx
x
x
x
x
0
4
2
3
0
4
2
3
4
62 5
16 16
62 5 16
8
3
4
∫
∫
=
+
− −
(
)
=
+
−
−






.
.
π
π
0 0
4
2
3
4
62 5 16 4 8 4
4
3
4
4
62 5 320
3
20 000
=
+
−
−






=
( ) =
.
( ) ( )
( )
( )
.
,
π
π
π
3 3
20 944
ft lb
ft lb
⋅
≈
⋅
,