Question

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L=2 cm if one side of the rectangle lies on the base of the triangle.

Answer 1

max Area = sqrt(3)/2

Answer 2

On the side of the triangle shared with a side of the rectangle call x the distance from one corner of the triangle to the start of the base of the rectangle. Then the height of the rectangle is x*tan(60)=x*sqrt(3). So the area is (2-2x)*(x*sqrt(3))=2xsqrt(3)-2x^2sqrt(3). We take the derivative and set equal to 0.

2sqrt(3)-4sqrt(3)x=0
x=1/2

So the base is 1 and the height is sqrt(3)/2

Max area =sqrt(3)/2

Answer 3

Thank you both very much!