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Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs as shown in the figure.
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure above and label further as follows
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| Photo by Math Principles in Everyday Life |
Area of Rectangle = Length x Width
Next, we need to use similar triangles in order to solve for the value of y as follows
Substitute the value of y to the above equation, we have
Take the derivative on both sides of the equation with respect to x, we have
Set dA/dx = 0 because we want to get the maximum area of the rectangle inscribed in a right triangle as follows
Consider
Substitute the value of x to the above equation, we have
Therefore, the area of the largest rectangle is
