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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Answers (1)

Makenna Lin
Answered 2022-07-20 Author has 16 answers
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 3 2 2 and
a smaller one having height h 1 = L 3 2 2 - x , that are similar! so we can write the
proportion L y = L 3 2 2 L 3 2 2 - x . By insulating the y we obtain y = L - 2 3 2 x
The rectangle area is S ( x , y ) = x y but
S ( x ) = x ( L - 2 3 2 x ) = L x - 2 3 2 x 2
By deriving S(x) we get S ( x ) = L - 4 3 2 x whose root is x = L 3 2 4 and
consequently y = L - 2 3 2 3 2 4 L = L 2
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