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.

Use Heron's formula A²=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

A² = (n+x/2)(x/2)(x/2)(n-x/2)
f(x) = 16A² = (2n+x)(x)(x)(2x-x)
f(x) = -x^4 + 4n²x²
f(x) has a critical point when f'(x)=0
f'(x) = -4x³ + 8nx
-4x³ + 8n²x = 0
-4x² + 8n² = 0
x² = 2n²
x = n√2

Posted by Jer on 2011-06-16 15:02:15