How do you find the rectangle of maximum area that can be inscribed in a right triangle with legs of length 3 and 4 if the sides of the rectangle are parallel to the legs of the triangle?
We write the area as a function of the width of the rectangle, which turns out to be quadratic which we maximize by completing the square, giving width
Let's pin our right triangle on the Cartesian plane with vertices
We'll place one corner of the rectangle at the origin as well, and sit the rectangle on the x axis so we have another corner at
The hypotenuse is a line through
So we have a rectangle with width
We could get the maximum with calculus, but this is asked as an algebra question so we complete the square. First we take out the factor on
Then we halve the coefficient on