2009 AMC 12A Problem 13

A ship sails $10$ miles in a straight line from $A$ to $B$ , turns through an angle between $45^{\circ}$ and $60^{\circ}$ , and then sails another $20$ miles to $C$ . Let $AC$ be measured in miles. Which of the following intervals contains $AC^2$ ?

$\textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800] \qquad \textbf{(E)}\ [800,900]$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.