2006 AMC 12B Problems/Problem 24

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Problem

Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$ . What is the area of the subset of $S$ for which

$\sin^2x-\sin x \sin y + \sin^2y \le \frac34?$

$\mathrm{(A)}\ \dfrac{\pi^2}{9}\qquad\mathrm{(B)}\ \dfrac{\pi^2}{8}\qquad\mathrm{(C)}\ \dfrac{\pi^2}{6}\qquad\mathrm{(D)}\ \df...$

Solution

We start out by solving the equality first. $\begin{align*}\sin^2x - \sin x \sin y + \sin^2y &= \frac34 \\\sin x &= \frac{\sin y \pm \sqrt{\sin^2 y - 4 ( \sin^2y ...$ We end up with three lines that matter: $x = y + \frac\pi3$ , $x = y - \frac\pi3$ , and $x = \pi - (y + \frac\pi3) = \frac{2\pi}{3} - y$ . We plot these lines below.

Note that by testing the point $(\pi/6,\pi/6)$ , we can see that we want the area of the pentagon. We can calculate that by calculating the area of the sqaure and then subtracting the area of the 3 triangles.

$\begin{align*}A &= \left(\frac{\pi}{2}\right)^2 - 2 \cdot \frac12 \cdot \left(\frac{\pi}{6}\right)^2 - \frac12 \cdot \lef...$

2006 AMC 12B (Problems • Resources)
Preceded by Problem 23	Followed by Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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2006 AMC 12B Problems/Problem 24

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