please help me sap with this calculus proof with differentiation

Question

performing any differentiation with a calculator that can determine gradient functions, show that for an isosceles triangle of fixed perimeter to have a maximum are the triangle needs to be equilateral

Answer 1

let the 2 legs both be x, making the third side P-2x.

Draw an altitude through the vertex angle and use the Pythagorean theorem so that the altitude can be represented as √(x² - (1/2 P - x)² which simplifies down to √(Px-1/4 P²).

 

The area then is A = 1/2 (P-2x)(√(Px-1/4 P²))

Take the derivative.

A' = (1/2 P-x)* 1/2(Px-1/4 P²)^-1/2 *P + (Px-1/4 P²)^1/2 *(-1)

 

Set this equal to 0 and solve.

 

(1/4P² - 1/2 Px) / (Px-1/4 P²)^1/2 = (Px-1/4P²)^1/2      cross multiply

 

1/4 P² -1/2 Px = Px - 1/4 P²     solve for x

1/2P² = 3/2 Px

1/3 P = x.

 

So the maximum area is when x is 1/3 of the perimeter....which is used for the 2 legs. The third side is P-2x or P-2(1/3P) which simplifies to 1/3 P which is the same as the other 2 sides, making this an equilateral triangle.

 

Hope you could follow this....and was helpful.