12. (a)
Note that
x
2
y
5
1125, so
y
5
}
11
x
2
2
5
}
. Then
c
5
5(
x
2
1
4
xy
)
1
10
xy
5
5
x
2
1
30
xy
5
5
x
2
1
30
x
1
}
11
x
2
2
5
}
2
5
5
x
2
1
33,750
x
2
1
}
d
d
c
x
}
5
10
x
2
33,750
x
2
2
5
}
10(
x
3
2
x
2
3375)
}
The critical point occurs at
x
5
15. Since
}
d
d
c
x
}
,
0 for
0
,
x
,
15 and
}
d
d
c
x
}
.
0 for
x
.
15, the critical point
corresponds to the minimum cost. The values of
x
and
y
are
x
5
15 ft and
y
5
5 ft.
(b)
The material for the tank costs 5 dollars/sq ft and the
excavation charge is 10 dollars for each square foot of
the crosssectional area of one wall of the hole.
13.
Let
x
be the height in inches of the printed area. Then the
width of the printed area is
}
5
x
0
}
in. and the overall
dimensions are
x
1
8 in. by
}
5
x
0
}
1
4 in. The amount of
paper used is
A
(
x
)
5
(
x
1
8)
1
}
5
x
0
}
1
4
2
5
4
x
1
82
1
}
40
x
0
}
in
2
. Then
A
9
(
x
)
5
4
2
400
x
2
2
5
}
4(
x
2
2
x
2
100)
}
and the critical point
(for
x
.
0) occurs at
x
5
10. Since
A
9
(
x
)
,
0 for
0
,
x
,
10 and
A
9
(
x
)
.
0 for
x
.
10, the critical point
corresponds to the minimum amount of paper. Using
x
1
8 and
}
5
x
0
}
1
4 for
x
5
10, the overall dimensions are
18 in. high by 9 in. wide.
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 Spring '08
 GERMAN
 Critical Point, Trigraph

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