1977 Canadian MO Problems/Problem 5: Difference between revisions
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 ..." |
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the vertex <math>V</math> to this path? | the vertex <math>V</math> 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 == | == Solution == | ||
Latest revision as of 14:53, 7 October 2014
Problem
A right circular cone of base radius 
cm and slant height of 
cm is given. 
is a point on the circumference of
the base and the shortest path from around the cone is drawn (see diagram). What is the minimum distance from
the vertex 
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]](http://latex.artofproblemsolving.com/5/4/6/54697301ec9de5ecc6b7a53b3ea1f523bd03347b.png)
