# Difference between revisions of "2003 AMC 8 Problems/Problem 24"

## Problem

A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?

$[asy]size(150); pair X=origin, A=(-5,0), B=(5,0), C=(0,5); draw(Arc(X, 5, 180, 360)^^B--C); dot(X); label("X", X, NE); label("C", C, N); label("B", B, E); label("A", A, W); [/asy]$

## Solution

The distance from $\text{X}$ to any point on the semicircle will always be constant. On the graph, this will represent a straight line. The distance between $\text{X}$ and line $\text{BC}$ will not be constant though. We can easily prove that the distance between $\text{X}$ and line $\text{BC}$ will represent a curve. As the ship travels from B to C, the distance between the ship and X will first decrease until it reaches the point Y so that XY is perpendicular to BC, and then increase afterwards. Hence the answer choice that fits them all is $\boxed{\text{(B)}}$.

## See Also

 2003 AMC 8 (Problems • Answer Key • Resources) Preceded byProblem 23 Followed byProblem 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 All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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