A rectangle is inscribed within a triangular region formed by the positive x-axis, the p...

A rectangle is inscribed within a triangular region formed by the positive x-axis, the positive y-axis, and the graph of the line x + y =50, as shown in the figure.

a. Determine the area as a function of the variable x.

b. Given the knowledge that there is a single value of x that produces a rectangle with maximum area, show that this rectangle must be a square.