Question

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side if one side of the rectangle lies on the base of the triangle. Answer ,

Answer 1

Let the length of base of rectangle be 'x'

Let the height of rectangle be 'y'

Thus we have for rectangle

`Area=Length* Breadth\\\\Area=x* y`

In the attached figure we can see that

`tan(60^(o))=(y)/(((a-x))/(2))\\\\\therefore y=(√(3))/(2)(a-x)\\\\Area=x* (√(3))/(2)(a-x)\\\\`

Differentiating area with respect to x and equating the result to zero we get

`A(x)=(√(3))/(2)(ax-x^(2))\\\\A'(x)=(√(3))/(2)(a-2x)=0\\\\\therefore x=(a)/(2)\\\\y=(√(3)a)/(4)`