1987 AIME Problems/Problem 4

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Problem

Find the area of the region enclosed by the graph of $\displaystyle |x-60|+|y|=|x/4|.$

Solution


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Since $|y|$ is nonnegative, $|\frac{x}{4}| \ge |x - 60|$. Solving this gives us two equations: $\frac{x}{4} \ge x - 60\ and \ -\frac{x}{4} \le x - 60$. Thus, $48 \le x \le 80$. The maximum and minimum y value is when $|x - 60| = 0$, which is when $x = 60$ and $y = \pm 15$. The area of the region enclosed by the graph is that of the quadrilateral defined by the points $(48,0),\ (60,15),\ (80,0), \ (60,-15)$. Breaking it up into triangles and solving, we get $2 \cdot \frac{1}{2}(80 - 48)(15 - (-15)) = 480$.

See also

1987 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions