2003 AMC 8 Problems/Problem 24: Difference between revisions

Revision as of 00:22, 2 July 2023

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$", A, W); [/asy]$

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Solution

The distance from Island $\text{X}$ to any point on the semicircle will always be constant. On the graph, this will represent a straight line. The distance between Island $\text{X}$ 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 Island $X$ will first decrease until it reaches the point $Y$ so that $\overline{XY}$ is perpendicular to $\overline{BC}$ , and then increase afterwards. Hence the answer choice that fits them all is $\boxed{\text{(B)}}$ .

Video Solution

https://www.youtube.com/watch?v=ibhy_qcQXiw ~David

Video Solution 2

https://youtu.be/TlCTRKo5mQ4 Soo, DRMS, NM

2003 AMC 8 (Problems • Answer Key • 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
All AJHSME/AMC 8 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

@@ Line 22: / Line 22: @@
 ==Video Solution==
 https://www.youtube.com/watch?v=ibhy_qcQXiw  ~David
+==Video Solution 2==
+https://youtu.be/TlCTRKo5mQ4   Soo, DRMS, NM
 ==See Also==

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