How do you find the area of the largest rectangle that can be inscribed in a right traingle with legs of lengths 3 cm and 4cm if two sides of the rectangle lie along the legs?

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How do you find the area of the largest rectangle that can be inscribed in a right traingle with legs of lengths 3 cm and 4cm if two sides of the rectangle lie along the legs?

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Answer 1 · 2015-05-27T00:36:26

Given the diagram below:
enter image source here
The area of the rectangle is
$A (x) = (3 - x) (\frac{4}{3} x)$ $A(x) = (3-x)(4/3x)$

$= 4 x - \frac{4}{3} x^{2}$ $= 4x- 4/3x^2$

The maximum area will occur when the derivative of the Area is equal to zero

$\frac{d A}{d x} = 4 - \frac{8}{3} x = 0$ $(dA)/(dx) =4 -8/3x = 0$

$4 = \frac{8}{3} x$ $4 = 8/3x$

$x = \frac{3}{2}$ $x = 3/2$

The width of the rectangle will be
$3 - x = 3 - \frac{3}{2} = \frac{3}{2}$ $3-x = 3 - 3/2 = 3/2$
The height of the rectangle will be
$\frac{4}{3} x = \frac{4}{3} \cdot \frac{3}{2} = 2$ $4/3 x = 4/3 * 3/2 = 2$

The Area of the rectangle will be
$\frac{3}{2} \cdot 2 = 3$ $3/2 * 2 = 3$
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How do you find the area of the largest rectangle that can be inscribed in a right traingle with legs of lengths 3 cm and 4cm if two sides of the rectangle lie along the legs?

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