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Difference between revisions of "1977 Canadian MO Problems/Problem 5"

(Created page with "== Problem == A right circular cone of base radius <math>1</math> cm and slant height of <math>3</math> cm is given. <math>P</math> is a point on the circumference of the base ...")
 
m (→‎Problem)
 
Line 5: Line 5:
 
the vertex <math>V</math> to this path?
 
the vertex <math>V</math> to this path?
  
−
{figure}
+
 
 +
<asy>
 +
path p1=yscale(.25)*arc((0,0),1,0,180);
 +
path p2=yscale(.25)*arc((0,0),1,0,-180);
 +
path q1=shift(-.25,.4)*rotate(30)*xscale(.85)*p1;
 +
path q2=shift(-.25,.4)*rotate(30)*xscale(.85)*p2;
 +
draw(p2,black);draw(q2,black);
 +
draw(p1,dashed);draw(q1,dashed);
 +
draw((-1,0)--(-.5,2.4)--(1,0));
 +
MP("P",(-1,0),W);MP("V",(-.5,2.4),N);
 +
draw((-.2,2.5)--(1.2,.2),arrow=ArcArrow());
 +
draw((1.2,.2)--(-.2,2.5),arrow=ArcArrow());
 +
draw((0,0)--(1,0),arrow=ArcArrow());
 +
draw((1,0)--(0,0),arrow=ArcArrow());
 +
MP("1 cm",(.5,.04),S);MP("3 cm",(.5,1.35),NE);
 +
</asy>
  
 
== Solution ==
 
== Solution ==

Latest revision as of 15:53, 7 October 2014

Problem

A right circular cone of base radius $1$ cm and slant height of $3$ cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path?


[asy] path p1=yscale(.25)*arc((0,0),1,0,180); path p2=yscale(.25)*arc((0,0),1,0,-180); path q1=shift(-.25,.4)*rotate(30)*xscale(.85)*p1; path q2=shift(-.25,.4)*rotate(30)*xscale(.85)*p2; draw(p2,black);draw(q2,black); draw(p1,dashed);draw(q1,dashed); draw((-1,0)--(-.5,2.4)--(1,0)); MP("P",(-1,0),W);MP("V",(-.5,2.4),N); draw((-.2,2.5)--(1.2,.2),arrow=ArcArrow()); draw((1.2,.2)--(-.2,2.5),arrow=ArcArrow()); draw((0,0)--(1,0),arrow=ArcArrow()); draw((1,0)--(0,0),arrow=ArcArrow()); MP("1 cm",(.5,.04),S);MP("3 cm",(.5,1.35),NE); [/asy]

Solution

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