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By all means (Posted on 2011-06-16) Difficulty: 3 of 5
In isosceles triangle AB=BC=n.

What value of AC warrants the largest area of the triangle ABC?
Solve by:
a) Plane geometry.
b) Trigonometry.
c) Calculus.
d) Any other way is welcome.

See The Solution Submitted by Ady TZIDON    
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A geometry + calculus way | Comment 2 of 5 |
Use Heron's formula A=s(s-a)(s-b)(s-c) where a,b,c are the sides and s=(a+b+c)/2

a=n, b=n, c=AB=x, s=n+x/2

A = (n+x/2)(x/2)(x/2)(n-x/2)
f(x) = 16A = (2n+x)(x)(x)(2x-x)
f(x) = -x^4 + 4nx
f(x) has a critical point when f'(x)=0
f'(x) = -4x + 8nx
-4x + 8nx = 0
-4x + 8n = 0
x = 2n
x = n√2
  Posted by Jer on 2011-06-16 15:02:15
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