Optimization: Find the Maximum Area

I need to find the rectangle of maximum area that can be inscribed in a right triangle with legs of length $a=53$ and $b=54$ if the sides of the rectangle $x,y$ are parallel to the legs of the triangle.

• Say one of the rectangle's corners lie in the right angle of the right angle triangle, and the opposite corner touches the hypotenuse. Then if one of the rectangle's sides (for instance the one along the leg $a$) is $x$ and the other side (along $b$) is $y$, then there is a very specific algebraic relation between $x$ and $y$. That relation is a mathematical formulation of the fact that one corner of the rectangle lies on the hypotenuse. Until you find that relation there is not much else you can do. – Arthur Nov 22 '15 at 16:48
• (i) First and most important step. Draw rhe triangle, with the side $a$ horizontal, the side $b$ vertical, and the right angle at the lower left. Now draw a candidate rectangle, Its upper right corner will be on the hypotenuse. Label two of the sides of the rectangle $x$ and $y$ in the obvious way. We want to maximize $xy$. (ii) Use the geometry to find a relationship between $x$ and $y$. Similar triangles will be useful. (iii) Use calculus machinery (or something else) to maximize. – André Nicolas Nov 22 '15 at 16:55

Maybee this image from mathalino.com will make you realize how to come up with the formula for the area of the rectangle in terms of $x$: Once you have the formula $f(x)=A$ find its maximum on the interval $[0,53]$ or $[0,54]$ depending on which leg you put on your horizontal $x$ axis. Consider $f'(x)=0$ or the endpoints of your interval.