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

(Created page with "The distance from X remains constant for the semicircle, so the first part of the graph has to be a straight line. Then, the line gets closer, and then farther away from X. So, (...")
 
 
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The distance from X remains constant for the semicircle, so the first part of the graph has to be a straight line. Then, the line gets closer, and then farther away from X. So, (A) would be the best option.
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==Problem==
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A ship travels from point <math>A</math> to point <math>B</math> along a semicircular path, centered at Island <math>X</math>. Then it travels along a straight path from <math>B</math> to <math>C</math>. Which of these graphs best shows the ship's distance from Island <math>X</math> as it moves along its course?
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<asy>size(150);
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pair X=origin, A=(-5,0), B=(5,0), C=(0,5);
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draw(Arc(X, 5, 180, 360)^^B--C);
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dot(X);
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label("$X$", X, NE);
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label("$C$", C, N);
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label("$B$", B, E);
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label("$A$", A, W);
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</asy>
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<center>
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[[Image:2003amc8prob24ans.png|800px]]
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</center>
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==Solution==
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The distance from Island <math>\text{X}</math> to any point on the semicircle will always be constant. On the graph, this will represent a straight line. The distance between Island <math>\text{X}</math> and line <math>\text{BC}</math> will not be constant though. We can easily prove that the distance between <math>\text{X}</math> and line <math>\text{BC}</math> will represent a curve. As the ship travels from <math>B</math> to <math>C</math>, the distance between the ship and Island <math>X</math> will first decrease until it reaches the point <math>Y</math> so that <math>\overline{XY}</math> is perpendicular to <math>\overline{BC}</math>, and then increase afterwards. Hence the answer choice that fits them all is <math>\boxed{\text{(B)}}</math>.
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==See Also==
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{{AMC8 box|year=2003|num-b=23|num-a=25}}
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[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 21:36, 9 October 2020

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]

2003amc8prob24ans.png

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 $\boxed{\text{(B)}}$.

See Also

2003 AMC 8 (ProblemsAnswer KeyResources)
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

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