 # 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? Lexi Mcneil 2022-07-19 Answered
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?
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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 $h=L\frac{\sqrt{3}}{2}$ and
a smaller one having height ${h}_{1}=L\frac{\sqrt{3}}{2}-x$, that are similar! so we can write the
proportion $\frac{L}{y}=\frac{L\frac{\sqrt{3}}{2}}{L\frac{\sqrt{3}}{2}-x}$. By insulating the y we obtain $y=L-\frac{2}{\sqrt{3}}x$
The rectangle area is $S\left(x,y\right)=x\cdot y$ but
$S\left(x\right)=x\cdot \left(L-\frac{2}{\sqrt{3}}x\right)=Lx-\frac{2}{\sqrt{3}}{x}^{2}$
By deriving S(x) we get $S\prime \left(x\right)=L-\frac{4}{\sqrt{3}}x$ whose root is $x=L\frac{\sqrt{3}}{4}$ and
consequently $y=L-\frac{2}{\sqrt{3}}\frac{\sqrt{3}}{4}L=\frac{L}{2}$