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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 more calculus way | Comment 3 of 5 |
call AC  = 2x
With AC as the base of the isosceles triangle the height can be found by the Pythagorean theorem. x+ h = n so h = √(n-x)

Area is then .5*2x*√(n-x)
A(x) = x√(n-x)

dA/dx = √(n-x) + x*.5*(n-x)^-.5*-2x
= √(n-x) -x/√(n-x)
= ((n-x)-x)/√(n-x)
=(n-2x)/√(n-x)

set this equal to zero means

n-2x = 0
x = n/2
x = n/√2 = n√(2)/2
AC = 2x = n√2
  Posted by Jer on 2011-06-16 15:17:30
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