Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?8 cms9 cms12 cmsNone of these

The area of a triangle is given by Heron's formula:

Area = sqrt[s(s - a)(s - b)(s - c)]

where s is the semi-perimeter of the triangle, and a, b, and c are the sides of the triangle.

In this case, a = 6, b = 10, and c = x.

The semi-perimeter s = (a + b + c) / 2 = (6 + 10 + x) / 2 = (16 + x) / 2 = 8 + x/2.

Substituting these values into Heron's formula gives:

Area = sqrt[(8 + x/2)(8 + x/2 - 6)(8 + x/2 - 10)(8 + x/2 - x)]

Simplifying this gives:

Area = sqrt[(8 + x/2)(2 + x/2)(-2 + x/2)(-x + x/2)]

The area of the triangle will be maximum when the value inside the square root is maximum.

This is a cubic function of x, and its maximum value can be found by taking the derivative and setting it equal to zero.

Doing this gives:

0 = 3x^2/4 - 16x + 32

Solving this quadratic equation gives:

x = [16 ± sqrt((16)^2 - 4332)] / (2*3)

x = [16 ± sqrt(256 - 384)] / 6

x = [16 ± sqrt(-128)] / 6

Since the square root of a negative number is not a real number, there are no real solutions to this equation.

Therefore, the area of the triangle cannot be maximized by choosing a specific value of x.

So, the correct answer is "None of these".