Consider the region in the \(x\)-\(y\) plane that is bounded by the \(x\)-axis and the function \(f(x) = 25-x^2\text{.}\) Construct a rectangle whose base lies on the \(x\)-axis and is centered at the origin, and whose sides extend vertically until they intersect the curve \(y = 25-x^2\text{.}\) Which such rectangle has the maximum possible area? Which such rectangle has the greatest perimeter?
