Chapter 5 Math And Sports Worksheet With Answers - The Mathematical Collage Page 6
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25The Mathematical Collage
§5.1 Scoring and Statistics
x (3 points)
y (1 point)
0
17
1
14
2
11
3
8
4
5
5
2
There are six ways in which the kicker could score 17 points.
Such an equation with integer coefficients and more that one integer solution is called a
Diophantine Equation after the Greek mathematician, Diophantus (c. 250 a.d.).
Example 5
An archery target consists of five concentric circles as shown. The value for an arrow in
each region starting from the inner circle is 9, 7, 5, 3, 1 points. In how many ways could
five scoring arrows earn 29 points?
Solution: We can set up Diophantine equations to
model the problem.
Let a = the number of 9 point arrows.
b = the number of 7 point arrows.
c = the number of 5 point arrows.
d = the number of 3 point arrows.
e = the number of 1 point arrows.
Since there are five scoring arrows,
+ + + + = 5.
a b c d e
Since the five arrows score 29 points,
9
7
5
3
1
29
.
a
b
c
d
e
+
+
+
+
=
We can now set up a table of values and systematically find all possible values that
simultaneously satisfy both equations. We can start with the largest value for a and
determine the values for the other variables making sure we use whole numbers and a
total of five arrows. The maximum value for a is 3, since when a reaches 4, 9a = 36,
which is over the 29 total points.
a (9)
b (7)
c (5)
d (3)
e (1)
3
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1
0
page 216
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