Find the dimensions of the rectangle with largest area that can be inscribed in an equilateral triangle with sides of 1 unit, if one side of the rectangle is on the base of the equilateral triangle.

The area of the rectangle is equal to L x W where L is the length and W is the width.

A = L x W

If we let L be the length and given that the triangle is equilateral with sides equal to 1, the value of W should be (0.5 - L/2) (tan 60)

W = (0.5 - L/2) (tan 60)

Simplifying the expression for the width,

W = 0.866 - 0.866L

The area now becomes,

A = (L) (0.866 - 0.866L)

Simplifying,

A = 0.866L - 0.866L²

Derive and equate to zero for the maximum value of L.

dA = 0.866 - 2 (0.866) (L) = 0

The value of L from the equation should be 0.5.

W = 0.866 - 0.866 (0.5) = 0.433

Answer: L = 0.5

W = 0.433