PUMaC 2012 · 几何（B 组） · 第 1 题

PUMaC 2012 — Geometry (Division B) — Problem 1

[ 3 ] During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle θ , in degrees, of the bottom of the cone when we look at it from the side?

解析

[ 3 ] During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle θ , in degrees, of the bottom of the cone when we look at it from the side? Solution: Let r be the radius of the original circle of the paper. Since we are opening up the second fold, the opening is made from the semicircle from the first fold. Thus the perimeter of the opening is πr . Therefore the radius of is opening is r/ 2. Draw a right triangle with one side as the altitude from the vertex to the opening, another side as the radius of the opening, and the third side as a slant, which is the radius of the original paper. Considering the angle between the slant and the altitude, we have that the hypoteneuse is double the opposite. Thus this is a 30-60-90 triangle, with the angle between the slant and the altitude equal to 30 degrees. Hence when we look at the cone from the side, half of the bottom angle is 30 degrees, and so the total angle is 60 degrees. Problem contributed by Xufan Zhang