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21.
If the upper right vertex of the rectangle is located at
(
x
, 4 cos 0.5
x
) for 0
,
x
,
p
, then the rectangle has width
2
x
and height 4 cos 0.5
x
, so the area is
A
(
x
)
5
8
x
cos 0.5
x
.
Then
A
9
(
x
)
5
8
x
(
2
0.5 sin 0.5
x
)
1
8(cos 0.5
x
)(1)
52
4
x
sin 0.5
x
1
8 cos 0.5
x
.
Solving
A
9
(
x
) graphically for 0
,
x
,
p
, we find that
x
<
1.72. Evaluating 2
x
and 4 cos 0.5
x
for
x
<
1.72, the
dimensions of the rectangle are approximately
3.44 (width) by 2.61 (height), and the maximum area is
approximately 8.98.
22.
Let the radius of the cylinder be
r
cm, 0
,
r
,
10. Then
the height is 2
ˇ
1
w
0
w
0
w
2
w
r
w
2
w
and the volume is
V
(
r
)
5
2
p
r
2
ˇ
1
w
0
w
0
w
2
w
r
w
2
w
cm
3
. Then
V
9
(
r
)
5
2
p
r
2
1
}
2
ˇ
1
w
0
w
1
0
w
2
w
r
w
2
w
}
2
(
2
2
r
)
1
(2
p
ˇ
1
w
0
w
0
w
2
w
r
w
2
w
)(2
r
)
5
5
The critical point for 0
,
r
,
10 occurs at
r
5
!
}
2
§
0
3
§
0
}
§
5
10
!
}
2
3
}
§
. Since
V
9
(
r
)
.
0 for
0
,
r
,
10
!
}
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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