Tandrew Ao, who came from the parallel universe, gives Andrew Tao an unusual gift – a cylinder that constantly changes its radius and height but has the same volume ($400 \pi$ cubic inches). Andrew Tao can carry it only in a rectangular box container made of Aonium; otherwise the cylinder will explode into a burst of $\gamma$-rays. Aonium is expensive and therefore Andrew Tao wants to use as little Aonium as possible. If a cubic inch of Aonium is $\$1\$, the Aonium container has to be at least half an inch thick, and the Aonium is sold only in cubic inches, what is the least amount of money that Andrew Tao has to spend to bring his gift back home?