The greatest product possible for one vertex in c) is
8
!
4
!
3
=
96
.
4
5
3
2
8
1
3
1
7

E
XTRA
C
HALLENGES
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2
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Faculty of Mathematics
University of Waterloo
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Thus, c. will yield the greatest product at one vertex.
4.
Let
OJ
represent the volume of orange juice,
PJ
represent the volume of pineapple juice,
CJ
represent the volume of cranberry juice,
W
represent the volume of water, and
V
represent the
total volume in the pitcher.
We know
OJ
+
PJ
+
CJ
+
W
=
V
.
Now,
2
CJ
=
PJ
,
OJ
=
PJ
and
W
=
1
8
V
.
So,
PJ
+
2
CJ
+
CJ
+
1
8
V
=
V
2
CJ
+
2
CJ
+
CJ
+
1
8
V
=
V
5
CJ
=
7
8
V
CJ
=
7
40
V
7
40
of the total volume is cranberry juice, so in 320 mL there is
7
40
!
320
or 56 mL of
cranberry juice.
5.
The area of the large square is 36 m
2
.
The diameter of the circle is equal to the length of one side of the large square, so the radius of
the circle is 3 m. Thus, the area of the circle is
9
!
m
2
.
The diagonal of the small square is equal to the diameter of the circle. The diagonal divides the
small square into two congruent right-angled isosceles triangles. Using the Pythagorean theorem,
we can determine the length of each side of the small square. If each side of the smaller square is
of length
a
, then
a
2
+
a
2
=
6
2
2
a
2
=
36
a
2
=
18
Thus, the area of the small square is 18 m
2
.
The area of the shaded region is
36
!
9
"
+
18
=
54
!
9
"
!
25.73
m
2
.

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