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

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==Solution==
 
==Solution==
  
The distance from <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 <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 semicircle (prove this by dividing <math>\triangle{XCB}</math> into two congruent triangles using the perpendicular bisector from vertex <math>\text{X}</math>). Since the point on  line <math>\text{BC}</math> and the perpendicular bisector from vertex <math>\text{X}</math> is the shortest distance between <math>\text{X}</math> and <math>\text{BC}</math> as well as the midpoint of line <math>\text{BC}</math> it will represent the shortest point on the semicircle in the graph as well as the midpoint of the semicircle. Using the information found, the answer choice that fits them all is <math>\boxed{\text{(B)}}</math>.
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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>.
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2003|num-b=23|num-a=25}}
 
{{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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