A cone with base radius and vertical height both equal to 10cm has been left on a horizontal table with its vertex pointing upwards by an absent-minded maths teacher.
Freddy the fly lands on the cone at a point exactly 5cm vertically above the table top. He then follows the shortest possible route across the curved surface of the cone to the point on the circumference of the base of the cone which is furthest from where he landed.
How long is Freddy’s route across the surface of the cone?
The conical surface can be flattened into a plane without distorting its geometry. The given cone has a slant height of 10*sqrt(2) and so its base will be an arc of that radius, but will occupy only 20*pi of the 10*sqrt(2)*2*pi circumference of a full circle of such radius.
Suppose we flatten the cone by cutting along a line from the vertex of the cone to Freddy's destination on the circumference of the base.
Let's consider the new location of the cone's apex to be the origin and Freddy's location on the x axis halfway to the circumference, and so at (5*sqrt(2),0). The destination is at either of the points on the circumference that are rotated 10*pi/(10*sqrt(2)) = pi/sqrt(2) radians from the positive x axis.
That puts the destination at (-8.565889667504388, 11.252801171448004),
which is 19.264993470431353 cm from Freddy's position at (7.071067811865476, 0).
|
|
Posted by Charlie
on 2024-08-04 08:48:39 |
solution
