Problem $1(5$ points$)$: A box with a square base and no top is constructed to have a volume of $500c{m}^3.$ Find the dimensions of the box that minimize the surface area.

Problem $2(5$ points$)$: A rectangle is inscribed in the parabola $y=108-x^2$ with its base on the $x-$axis and its two upper corners on the parabola. Find the dimensions that maximize the area of the rectangle. Draw the pictures