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

 
(4 intermediate revisions by 2 users not shown)
Line 18: Line 18:
 
==Solution==
 
==Solution==
  
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 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 <math>\boxed{\text{(B)}}</math>.
+
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>.
 +
 
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=ibhy_qcQXiw  ~David
 +
 
 +
==Video Solution 2==
 +
https://youtu.be/TlCTRKo5mQ4  Soo, DRMS, NM
  
 
==See Also==
 
==See Also==

Latest revision as of 01:22, 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]

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

Video Solution

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

Video Solution 2

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

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

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