How do you find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle?

1 Answer
Nov 2, 2016

base=L/2 height=L*root2{3}/4

Explanation:

Let the upper base y of the rectangle be the segment of a line parallel to the base of the equilateral triangle at an unknown distance x from it. In such a way the triangle is divided in two triangles, the equilateral one having height h=Lroot2{3}/2 and a smaller one having height h_1=Lroot2{3}/2-x, that are similar! so we can write the proportion \frac{L}{y}=\frac{Lroot2{3}/2}{Lroot2{3}/2-x}. By insulating the y we obtain y=L-\frac{2}{root2{3}}x

The rectangle area is S(x,y)=x*y but S(x)=x*(L-\frac{2}{root2{3}}x)=Lx-\frac{2}{root2{3}}x^2.

By deriving S(x) we get S'(x)=L-\frac{4}{root2{3}}x whose root is x=L\frac{root2{3}}{4} and consequently y=L-\frac{2}{root2{3}}\frac{root2{3}}{4}L=\frac{L}{2}