Bunuel wrote:

A semicircular piece of paper has center O, as shown above. Its diameter A’A is coated with adhesive. If the adhesive is used to fuse radii OA’ and OA along their entire lengths (so that points A and A’ coincide, points P and P’ coincide, and so on), a cone is formed as shown above. If point B divides the original semicircle into two identical arcs, what is the measure of angle AOB in the folded cone?

A. 45º

B. Between 45º and 60º

C. 60º

D. Between 60º and 90º

E. 90º

Attachment:

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Kudos for a correct solution. MANHATTAN GMAT OFFICIAL SOLUTION:If you’re having trouble visualizing the relationship between the flat piece of paper and the 3D cone, get out your scissors and make a cone yourself. (You can’t do this on the real test, of course, but this is a challenge problem—you probably wouldn’t see a problem this hard on the real test!)

There are two key relationships between the semicircle (the flat piece of paper) and the cone (the 3D shape). First, the radius of the semicircle becomes the slant height of the cone. The slant height is the length measured along the outer surface of the cone, from A’ to O in the diagram). Second, the “half-circumference” of the semicircle (i.e., the length just of the rounded part, not including the diameter) becomes the circumference of the cone’s base (the circle on the bottom of the cone).

Let R stand for the radius of the original semicircle. If there were a full circle the circumference would be 2πR, so the “half-circumference” (through points A, B, and A’) is half of that, or just πR. This length becomes the full circumference of the cone’s circular base.

Find the radius, r, of the cone’s circular base:

2πr = πR

r = R/2

The diameter of the cone’s base is thus 2(R/2) = R.

As observed above, the cone’s slant height is also R, since it’s the same as the radius OA or OA’ of the original semicircle.

Now consider the triangle AOB. The two sides OA and OB are both equal to the cone’s slant height, so each is R units. The bottom side AB is a diameter of the cone’s base, so its length is also R units.

The triangle AOB is equilateral! The measure of each angle must be 60°, so the measure of angle AOB is 60°.

The correct answer is (C).

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