A rectangle has its base on the $x$-axis and two vertices on the curve $y=100-4x^2$. Determine the length $l$ and width $w$ that yield the maximum area. Then, find the maximum area.

$l=\frac{200}{3}\text{ units}$

$w=\frac{10\sqrt{3}}{3}\text{ units}$

$A_{\max }=\frac{2000\sqrt{3}}{9}\text{ square units}$

$l=\frac{200}{3}\text{ units}$

$w=\frac{5\sqrt{3}}{3}\text{ units}$

$A_{\max }=\frac{1000\sqrt{3}}{9}\text{ square units}$

$l=\frac{100}{3}\text{ units}$

$w=\frac{10\sqrt{3}}{3}\text{ units}$

$A_{\max }=\frac{1000\sqrt{3}}{9}\text{ square units}$

$l=\frac{100}{3}\text{ units}$

$w=\frac{5\sqrt{3}}{3}\text{ units}$

$A_{\max }=\frac{500\sqrt{3}}{9}\text{ square units}$