Question

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 $AC^2?$

$[400, 500]$

$[500, 600]$

$[600, 700]$

$[700, 800]$

$[800, 900]$

Answer: D

Concepts:law of cosines bounding to limit cases

Difficulty rating: 1770

Solution:

By the Law of Cosines, $\begin{aligned} AC^2 &= 10^2 + 20^2 \\ &\quad {}- 2\cdot 10\cdot 20\cos(\angle ABC) \\ &= 500 - 400\cos(\angle ABC). \end{aligned}$

The ship turns through an angle between $45^\circ$ and $60^\circ,$ so the interior angle $\angle ABC$ lies between $120^\circ$ and $135^\circ.$

Since $\cos 120^\circ = -\dfrac{1}{2}$ and $\cos 135^\circ = -\dfrac{\sqrt{2}}{2},$ $\begin{aligned} 700 &= 500 + 200 \\ &\le AC^2 \le 500 + 200\sqrt{2} \\ &\lt 800. \end{aligned}$

So $AC^2$ lies in $[700, 800].$

Thus, the correct answer is D.

2009 AMC 12A Problem 13

Answer: D

Difficulty rating: 1770

Solution:

Problem 13 in Other Years