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)]

To find the maximum area, we need to find the value of x that maximizes this expression.

Taking the derivative of the area with respect to x and setting it equal to zero gives:

dArea/dx = 0

Solving this equation for x gives the value of x that maximizes the area.

However, this is a complex calculation that requires knowledge of calculus.

A simpler approach is to note that for a given perimeter, a triangle has maximum area when it is equilateral.

In this case, the perimeter is 16 + x, so the maximum area occurs when all sides are equal, i.e., when x = 16 + x / 3.

Solving this equation for x gives x = 8.

Therefore, the maximum area of the triangle occurs when x = 8 cm.