A right circular cone of base radius 17cm and slant height 51cm is given. P is a point on the circumference of the base. Find the length of the shortest path from P around the cone and back to P.
If you cut the cone from P to the vertex, then flatten it, you get a section of a circle. This circle has radius 51, the slant height. This circular section is exactly 1/3 of a circle since the radius (and therefore circumference) of the cone (17) is 1/3 the radius of our circular section.
In the flattened cone, the shortest path from P to P' is a straight line, a chord of a 120 degree arc of a circle whose radius is 51.
We have a 30-30-120 triangle whose sides are 51, 51, and 51√3 (about 88.33).
This pathway comes closest to the apex of the cone 51/2 cm from the apex when it is on the opposite side of the cone from P.
Shortest path: 51√3 (about 88.33)
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Posted by Larry
on 2024-11-05 08:44:47 |
Solution
