The lengths of the sides of a triangle are a+x^2 , a+x^2 and 2a-2x^2 , where a > 0 is a constant and x is a real variable. If the maximum possible area of this triangle is 36 3 , then the value of a is

Step-by-step solution

Let the sides of the triangle be A = a+x^2 , B = a+x^2 , and C = 2a-2x^2 . The semi-perimeter s is given by: s = A+B+C 2 = a+x^2 + a+x^2 + 2a-2x^2 2 = 2a Using Heron's formula, the area of the triangle is: = s(s-A)(s-B)(s-C) = 2a(2a - a - x^2)(2a - a - x^2)(2a - 2a + 2x^2) = 2a(a-x^2)(a-x^2)(2x^2) = 4ax^2(a-x^2)^2 = 2 a x(a-x^2) To find the maximum area, we maximize the function f(x) = x(a-x^2) = ax - x^3 . Differentiating with respect to x and setting it to zero: f'(x) = a - 3x^2 = 0 x^2 = a 3 Substituting x^2 = a