$10.$ A red$-$tailed hawk is $180$ m above the ground, flying east at $\backslash (15\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$sec$\}\backslash ),$ when it drops its prey. Before it hits the ground, the prey's falling path is given by the curve $\backslash ($ y$=180-\backslash$frac$\{$x$^\{2\}\}\{45\}\backslash ),$ where $\backslash ($ y $\backslash )$ is the prey's height above the ground, and $\backslash ($ x $\backslash )$ is the horizontal distance traveled by the prey before it landed.

$($a$)$ Give a sketch that describes this situation, including the prey's trajectory as it falls.

$($b$)$ How far east does the prey hit the ground, relative to the location where the hawk dropped it$?$ Include calculations that justify your reasoning.

$($c$)$ Calculate the length of the prey's falling path from the time it is dropped until the time it hits the ground. Express your answer to the nearest tenth of a meter.

In each step explain the step and why.