Problems(1)
Which of the cones listed below can be formed from a 252∘ sector of a circle of radius 10 by aligning the two straight sides?
[asy]import graph;unitsize(1.5cm);defaultpen(fontsize(8pt));draw(Arc((0,0),1,-72,180),linewidth(.8pt));draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));label("10",(-0.5,0),S);draw(Arc((0,0),0.1,-72,180));label("252∘",(0.05,0.05),NE);[/asy]
[asy]
import three;
picture mainframe;
defaultpen(fontsize(11pt));
picture conePic(picture pic, real r, real h, real sh)
{
size(pic, 3cm);
triple eye = (11, 0, 5);
currentprojection = perspective(eye);
real R = 1, y = 2;
triple center = (0, 0, 0);
triple radPt = (0, R, 0);
triple negRadPt = (0, -R, 0);
triple heightPt = (0, 0, y);
draw(pic, arc(center, radPt, negRadPt, heightPt, CW));
draw(pic, arc(center, radPt, negRadPt, heightPt, CCW), linetype("8 8"));
draw(pic, center--radPt, linetype("8 8"));
draw(pic, center--heightPt, linetype("8 8"));
draw(pic, negRadPt--heightPt--radPt); label(pic, (string) r, center--radPt, dir(270));
if (h != 0)
{
label(pic, (string) h, heightPt--center, dir(0));
}
if (sh != 0)
{
label(pic, (string) sh, heightPt--radPt, dir(0));
}
return pic;
}
picture pic1;
pic1 = conePic(pic1, 6, 0, 10);
picture pic2;
pic2 = conePic(pic2, 6, 10, 0);
picture pic3;
pic3 = conePic(pic3, 7, 0, 10);
picture pic4;
pic4 = conePic(pic4, 7, 10, 0);
picture pic5;
pic5 = conePic(pic5, 8, 0, 10);picture aux1; picture aux2; picture aux3;
add(aux1, pic1.fit(), (0,0), W);
label(aux1, "<spanclass=′latex−bold′>(A)</span>", (0,0), 22W, linewidth(4));
label(aux1, "<spanclass=′latex−bold′>(B)</span>", (0,0), 3E);
add(aux1, pic2.fit(), (0,0), 35E);add(aux2, aux1.fit(), (0,0), W);
label(aux2, "<spanclass=′latex−bold′>(C)</span>", (0,0), 3E);
add(aux2, pic3.fit(), (0,0), 35E);add(aux3, aux2.fit(), (0,0), W);
label(aux3, "<spanclass=′latex−bold′>(D)</span>", (0,0), 3E);
add(aux3, pic4.fit(), (0,0), 35E);add(mainframe, aux3.fit(), (0,0), W);
label(mainframe, "<spanclass=′latex−bold′>(E)</span>", (0,0), 3E);
add(mainframe, pic5.fit(), (0,0), 35E);
add(mainframe.fit(), (0,0), N);
[/asy] geometry3D geometryAMC