Question

the hypotenuse of a right triangle is 12 cm. find one the legs if the area is maximum?

Answer 1

Intuitively, one should conclude that the sides would be equal, let's prove it

let one side by x, then the other is √(144 - x^2)

area = (1/2)x√(144-x^2) = (1/2)x(144-x^2)^(1/2)
d(area)/dx
= (1/2)[ x(1/2)(144-x^2)^(-1/2) (-2x) + (144-x^2)^(1/2) ]
= 0 for a max of area

-x^2/√(144-x^2) + √(144-x^2) = 0
x^2/√(144-x^2) = √(144-x^2)
x^2 = 144-x^2
2x^2 = 144
x^2 = 72
x = √72 = 6√2
one side is 6√2 , and the other side is
√(144 - 72) = √72 = 6√2
as anticipated.

check:
(√72)^2 + (√72)^2
= 144

Answer 2

tnx