Mathematics

Mathematics

Problem

A rectangle is inscribed in an equilateral triangle of side length 2a2a units. The maximum area of this rectangle can be (a) 3a2\sqrt{3} a^2 (b) 3a24\frac{\sqrt{3} a^2}{4} (c) a2a^2 (d) 3a22\frac{\sqrt{3} a^2}{2}
Intuitive explanation

The key is to set up a coordinate system with the triangle's base on the xx-axis and use the constraint that the rectangle's top two vertices must lie on the slanted sides of the triangle. Express the rectangle's width and height in terms of a single variable (the height of the rectangle), then optimize using calculus. The interplay between increasing height and decreasing width will reveal where the area peaks.

Intuitive explanation

The key is to set up a coordinate system with the triangle's base on the xx-axis and use the constraint that the rectangle's top two vertices must lie on the slanted sides of the triangle. Express the rectangle's width and height in terms of a single variable (the height of the rectangle), then optimize using calculus. The interplay between increasing height and decreasing width will reveal where the area peaks.

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