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I was given with the question below:

A circular cone with vertex C has a point A on the circumference of its base and
point B on the segment AC, as shown in the diagram below. The shortest
possible rope is wrapped once around the cone such that it starts at point A and
ends at point B. Assume that the base diameter and the slant height of this cone
are 6 cm and 12 cm, respectively. If AB = 3 cm and D is the point on the rope
that is closest to C, what is the length, in cm, of CD?

With this diagram

Instinctively, by just looking at the diagram, I believe that B is the point on the rope that is closest to C. Therefore D is the same as B, and CD = 12 - 3 = 9.

However, this looks to simple to be true, and it doesn't make much sense to have two points overlap. Furthermore, I found this question from this website, where the 15th question of EMIC is exactly the question I was tasked with, and the solution claims that it is 7.2.

The value of 7.2 makes no sense to me, so I tried to attempt the question again, by modelling the rope as a helix with parametric equation

$$ x(t) = 3(1-t)\cos(8\pi t)\\ y(t) = 3(1-t)\sin(8\pi t)\\ z(t) = 3\sqrt{15}t\\ 0<t\leq\frac{1}{4}\\ $$

which, when plotted in desmos, seems to be exactly the rope I was given with in the question. I then attempted to use the distance formula to calculate the distance between a point on the parametric equation above, and the point $(0, 0, 3\sqrt{15})$ (which is the vertex of the cone, calculated by the Pythagorean theorem). The result was that, for $t=t$, the distance between the rope and the vertex is

$$ 12|1-t| $$

Obviously, to minimise $12|1-t|$ for $0<t\leq\frac{1}{4}$, $t=\frac{1}{4}$ when the distance becomes 9, validating my initial guess.

My answer should be wrong, but I just can't spot any mistake that I have made. I would appreciate if you could find my mistake and provide some hints to the correct solution.

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    $\begingroup$ The reason your answer is wrong is that you assumed that the shortest path is a helix-like curve. Why should the $z$ be proportional to $t$? It would make more sense to start with a bigger slope at the beginning, followed by a smaller one, when radius is smaller. $\endgroup$
    – Andrei
    Feb 12 at 9:35

1 Answer 1

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The trick is to unwrap the side of the cone. Since the side is $AC=12$, you get a circular sector with the same radius. To find the angle, the length is $2\pi \frac 62=6\pi$. Then the angle of the circular sector is then $\frac{6\pi}{12}=\frac{\pi}2$. So point $A$ is at a distance $12$ from $C$, point $B$ is at a distance $9$ from $C$. See the figure below:

enter image description here

It should be obvious that the shortest path for the rope is the $AB$ line. You notice that the closest point to $C$ is the foot of the perpendicular from $C$ to $AB$. Since $\triangle ABC$ is a right angle triangle, we can write area in two ways:$$\frac12 AB\cdot CD=\frac12 CB\cdot CA$$ Can you take it from here?

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