SOLUTION: flag poles A and B are both perpendicular to the ground. Wires from the top of one flagpole are connected to the bottom of the other flagpole such that the point where the wires in

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Question 1151077: flag poles A and B are both perpendicular to the ground. Wires from the top of one flagpole are connected to the bottom of the other flagpole such that the point where the wires intersect is 10 feet above the ground. If the height of both A and B are an integer number of feet, how many possible heights are there for A?
Found 2 solutions by greenestamps, Alan3354:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The basic problem here is a common one. With the two poles of heights a and b, the height above the ground at which the two wires cross is ab%2F%28a%2Bb%29

So in this problem we have

ab%2F%28a%2Bb%29=10

and we need to determine the number of possible values for a if a and b are both integers.

This is a Diophantine equation -- we have two variables but only one equation; but the number of solutions is limited by the requirement that both variables be integers.

The following is one standard way for solving such equations.

(1) Solve the equation for one variable in terms of the other.

ab%2F%28a%2Bb%29+=+10
ab+=+10a%2B10b
ab-10b+=+10a
b%28a-10%29+=+10a
b+=+10a%2F%28a-10%29
b+=+%2810a-100%2B100%29%2F%28a-10%29
b+=+%2810a-100%29%2F%28a-10%29%2B100%2F%28a-10%29
b+=+10%2B100%2Fa-10

(2) b and 10 are integers; that means 100%2F%28a-10%29 must be an integer. And that means (a-10) must be a factor of 100.

We could identify all the possible values of a; but the problem only asks for the number of possible values for a. 100 has 9 positive factors, so there are 9 possible values for a.

ANSWER: There are 9 possible heights for pole A.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
flag poles A and B are both perpendicular to the ground. Wires from the top of one flagpole are connected to the bottom of the other flagpole such that the point where the wires intersect is 10 feet above the ground. If the height of both A and B are an integer number of feet, how many possible heights are there for A?
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Note that the distance between them is not relevant.
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Also, the height of the intersection is a*b/(a + b) - you can work that out or take my word for it.
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a*b/(a+b) = 10
1 obvious solution is a = b = 20.
It's also obvious that the height of both must be greater than 10 feet.
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a*b = 10a + 10b
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By trial and error (using Excel) the combinations are:
20 & 20
30 & 15
35 & 14
60 & 12
110 & 11
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---> 10 possible values, but 20 is a duplicate.
---> 9 different possible values for A (or B).