SOLUTION: A right triangle has a fixed hypotenuse of 30 cm. And the other two sides are allowed to vary. Determine the largest possible area of the triangle.

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Question 934802: A right triangle has a fixed hypotenuse of 30 cm. And the other two sides are allowed to vary. Determine the largest possible area of the triangle.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
x= length of one leg, in cm.
y= length of the other leg, in cm.
}}}According to the Pythagorean theorem,
x%5E2%2By%5E2=30%5E2<--->x%5E2%2By%5E2=900<--->y%5E2=900-x%5E2
The area of the right triangle, in cm%5E2 is
area=%281%2F2%29%2Ax%2Ay,
so area%5E2=%28%281%2F2%29%2Ax%2Ay%29%5E2
area%5E2=%281%2F2%29%5E2%2Ax%5E2%2Ay%2A2
area%5E2=%281%2F4%29%2Ax%5E2%2Ay%2A2
Substituting the expression 900-x%5E2 (found above) for y%5E2 we get
area%5E2=%281%2F4%29%2Ax%5E2%2A%28900-x%5E2%29<---->area%5E2=%281%2F4%29%28-x%5E4%2B900x%5E2%29<---->area%5E2=%281%2F4%29%28-%28x%5E2%29%5E2%2B900x%5E2%29
That last expression shows area%5E2 a as quadratic function of x%5E2%29 ,
and we know that it will have a maximum when x%5E2=900%2F2<--->x%5E2=450 .
For that value of x%5E2 ,
area%5E2=%281%2F4%29%28-450%5E2%2B900%2A450%29
area%5E2=%281%2F4%29%2A450%2A%28-450%2B900%29
area%5E2=%281%2F4%29%2A450%2A450
area%5E2=%281%2F4%29%2A450%5E2
area%5E2=450%5E2%2F2%5E2
area%5E2=%28450%2F2%29%5E2
area%5E2=225%5E2
highlight%28area=225%29 and that is the largest possible area of the triangle, in square centimeters.