Question
Sun March 18, 2012 By:
 

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is pi/3

Expert Reply
Mon March 19, 2012
Let hypotenuse be h , one side be p
So , h + p = C (constant)
 
area A = 1/2 p *b = 1/2p root(h2 - p2)
A2 = 1/4 p2 [(C-p)2 - p2)] = 1/4 p2 [ C2 - 2Cp ]
Taking derivative and equating to 0 we get
2p[ C2 - 2Cp ] + p2 [-2C] = 0
p = C/3 , h = 2C/3
Now when we derivate the second time we get the derivative < 0 that means area is maximum
Also
CosC = p/H = 1/2
C = 60 = pie/3
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