Difference between revisions of "2008 AMC 12B Problems/Problem 20"

Revision as of 13:29, 26 November 2009

The following problem is from both the 2008 AMC 12B #20 and 2008 AMC 10B #25, so both problems redirect to this page.

Problem

Your mom walks at the rate of $10$ feet per second on a long straight path. Trash pails are located every $300$ feet along the path. A garbage truck travels at $15$ feet per second in the same direction as your mom and stops for $30$ seconds at each pail. As your mom passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will your mom and the truck meet?

$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8$

The solution is your mom.

2008 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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 AMC 12 Problems and Solutions

2008 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Question
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 AMC 10 Problems and Solutions

@@ Line 6: / Line 6: @@
 <math>\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 8</math>
-==Solution==
+The solution is your mom.
-Pick a coordinate system where Michael's starting pail is <math>0</math> and the one where the truck starts is <math>200</math>.
-Let <math>M(t)</math> and <math>T(t)</math> be the coordinates of Michael and the truck after <math>t</math> seconds.
-Let <math>D(t)=T(t)-M(t)</math> be their (signed) distance after <math>t</math> seconds.
-Meetings occur whenever <math>D(t)=0</math>.
-We have <math>D(0)=200</math>.
-The truck always moves for <math>20</math> seconds, then stands still for <math>30</math>. During the first <math>20</math> seconds of the cycle the truck moves by <math>200</math> meters and Michael by <math>100</math>, hence during the first <math>20</math> seconds of the cycle <math>D(t)</math> increases by <math>100</math>.
-During the remaining <math>30</math> seconds <math>D(t)</math> decreases by <math>150</math>.
-From this observation it is obvious that after four full cycles, i.e. at <math>t=200</math>, we will have <math>D(t)=0</math> for the first time.
-During the fifth cycle, <math>D(t)</math> will first grow from <math>0</math> to <math>100</math>, then fall from <math>100</math> to <math>-50</math>. Hence Michael overtakes the truck while it is standing at the pail.
-During the sixth cycle, <math>D(t)</math> will first grow from <math>-50</math> to <math>50</math>, then fall from <math>50</math> to <math>-100</math>. Hence the truck starts moving, overtakes Michael on their way to the next pail, and then Michael overtakes the truck while it is standing at the pail.
-During the seventh cycle, <math>D(t)</math> will first grow from <math>-100</math> to <math>0</math>, then fall from <math>0</math> to <math>-150</math>. Hence the truck meets Michael at the moment when it arrives to the next pail.
-Obviously, from this point on <math>D(t)</math> will always be negative, meaning that Michael is already too far ahead. Hence we found all <math>\boxed{5}</math> meetings.
-The movement of Michael and the truck is plotted below: Michael in blue, the truck in red.
-<asy>
-import graph;
-size(400,300,IgnoreAspect);
-real[] xt = new real[21];
-real[] yt = new real[21];
-for (int i=0; i<11; ++i) xt[2*i]=50*i;
-for (int i=0; i<10; ++i) xt[2*i+1]=50*i+20;
-for (int i=0; i<11; ++i) yt[2*i]=200*(i+1);
-for (int i=0; i<10; ++i) yt[2*i+1]=200*(i+2);
-real[] xm={0,500};
-real[] ym={0,2500};
-draw(graph(xt,yt),red);
-draw(graph(xm,ym),blue);
-xaxis("$time$",Bottom,LeftTicks);
-yaxis("$position$",Left,LeftTicks);
-</asy>
 ==See also==
 {{AMC12 box|year=2008|ab=B|num-b=19|num-a=21}}
 {{AMC10 box|year=2008|ab=B|num-b=24|after=Last Question}}

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Difference between revisions of "2008 AMC 12B Problems/Problem 20"

Revision as of 13:29, 26 November 2009

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