## Problem of the Day #66: Banner Across an Arch
*May 24, 2011*

*Posted by Albert in : potd , trackback*

After constructing a giant arch in the shape of one sinusoidal hump, Billy wants to place a banner across it to advertise mpotd.com. But the giant arch is… GIANT! (built to the standard $1$km high and $\frac{12\sqrt{2}}{5}$km wide), so Billy needs to figure out how he minimize cost while still making the banner visible from a certain distance. To minimize costs, Billy decides to hang the rectangular banner from two spots at equal hight on opposite sides of the arch, and have the banner touch the ground. He also remembers reading that a nominal area, $A$, for the banner is one such that $\frac{A}{d^2} = \frac{1}{64}$, where $d$ is the maximum viewing distance from the banner. Billy remembers that the arch was purposefully built a distance of $64(\sqrt{3}-1)$km from a mountain range (for whatever reason), so he figures this is the maximum distance needed. What length of banner paper should Billy buy?

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