SOLUTION: Ashley saved a distance equal to 80% of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two of the sides. Find t

Algebra ->  Length-and-distance -> SOLUTION: Ashley saved a distance equal to 80% of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two of the sides. Find t      Log On


   

Question 1073186: Ashley saved a distance equal to 80% of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two of the sides. Find the ratio of the length of the shortest side to the length of the longest.
Answer by josgarithmetic(39627) About Me  (Show Source):
You can put this solution on YOUR website!
x, longest side
y, shortest side
r, the diagonal path

Total path actually used, r=x%2By-0.8y
r=x%2B0.2y
r=x%2By%2F5
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Question means, to find the ratio y%2Fx.



r is also hypotenuse of the right triangle of sides x and y.
r%5E2=x%5E2%2By%5E2


Also from formula for r, r%5E2=%28x%2By%2F5%29%5E2=x%5E2%2B2xy%2F5%2By%5E2%2F25

The two expressions for r^2 are equal.
x%5E2%2By%5E2=x%5E2%2B2xy%2F5%2By%5E2%2F25
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Algebra steps
y%5E2=2xy%2F5%2By%5E2%2F25

25y%5E2=10xy%2By%5E2

Divide both sides by y%5E2.
25y%5E2%2Fy%5E2=10xy%2Fy%5E2%2By%5E2%2Fy%5E2
25=10x%2Fy%2B1
24=10x%2Fy
24%2F10=x%2Fy
y%2Fx=10%2F24
highlight%28y%2Fx=5%2F12%29, ratio of shortest side to longest side