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

2009 AMC 12A Problem 13 — Solution

2009 AMC 12A Problem 13

Below is the professionally curated solution for Problem 13 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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Concepts:law of cosines bounding to limit cases

Difficulty rating: 1770

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 $AC^2?$

$[400, 500]$

$[500, 600]$

$[600, 700]$

$[700, 800]$

$[800, 900]$

Solution:

By the Law of Cosines, $AC^2 = 10^2 + 20^2 - 2\cdot 10\cdot 20\cos(\angle ABC) = 500 - 400\cos(\angle ABC).$

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 $\cos 135^\circ = -\dfrac{\sqrt{2}}{2},$ $700 = 500 + 200 \le AC^2 \le 500 + 200\sqrt{2} \lt 800.$

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

Thus, the correct answer is D.

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