SOLUTION: A triangular garden in the shape of a right triangle has an area of 60m² If the longest side is 2 meters longer than the next shorter side which is x meters long, derive the pol

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: A triangular garden in the shape of a right triangle has an area of 60m² If the longest side is 2 meters longer than the next shorter side which is x meters long, derive the pol      Log On


   

Question 1178000: A triangular garden in the shape of a right triangle has an area of 60m²
If the longest side is 2 meters longer than the next shorter side which is x meters long, derive the polynomial equation x³+x²-3600=0 and then solve for x.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52750) About Me  (Show Source):
Answer by greenestamps(13195) About Me  (Show Source):
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The longest side of a right triangle is the hypotenuse. In this problem, it is 2 longer than the next longest side, which has length x.

So the right triangle has long leg x and hypotenuse x+2.

The Pythagorean Theorem then gives us sqrt(4x+4) = 2*sqrt(x+1) as the length of the other leg.

The area of a right triangle is one-half the product of the lengths of the two legs; the area is given as 60:

60+=+%281%2F2%29%28x%29%282%2Asqrt%28x%2B1%29%29+=+x%28sqrt%28x%2B1%29%29
3600+=+x%5E2%28x%2B1%29+=+x%5E3%2Bx%5E2
x%5E3%2Bx%5E2-3600+=+0

Solve using a graphing calculator to find x=15 as the only real root.

ANSWER: x=15

Note that gives us the familiar 8-15-17 right triangle....