Dimensions of the rectangle with largest area inscribed in an equilateral triangle (KristaKingMath)

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Learn how to find the dimensions of the rectangle that maximize its area, assuming the rectangle is inscribed in an equilateral triangle of side L. In order to complete this optimization problem, you'll need to draw a picture of the problem and write down everything you know, identify optimization and constraint equations, and then use the derivative of the optimization equation to find the dimensions. The optimization equation will be an equation for the area of the rectangle in terms of its base and height. If you draw the equilateral triangle and the rectangle on an xy coordinate plane with their bases on the x-axis and the figures symmetric about the y-axis, you can describe the right-edge of the base as extending to the coordinate (x,0). Therefore, x is equal to half of the base, or b/2. Then, use the method of similar triangles and the Pythagorean theorem to relate the base and height of the smaller triangle with the larger triangle. This allows you to solve for h. Plug the value you got for the height and half the base back into your area equation. Then simplify it, take its derivative, set the derivative equal to 0, and solve for x. This will lead you to the length of the base. Then you can plug this into your height equation to find the height of the rectangle.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

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