Problem Solving

A sphere is cut into two equal parts. Now out of two equal pieces, 1 piece is again cut into a symmetrical fashion such that 2 equal pieces are formed. Now, once again, one piece from newly formed pieces is taken and cut into a symmetrical fashion such that two equal pieces are formed. Then what will be the ratio of surface area of the biggest piece to the smallest piece ?

Options:
A) 3.6
B) 3.2
C) 1.8
D) 2.4
E) None of these

adthedaddy wrote:A sphere is cut into two equal parts. Now out of two equal pieces, 1 piece is again cut into a symmetrical fashion such that 2 equal pieces are formed. Now, once again, one piece from newly formed pieces is taken and cut into a symmetrical fashion such that two equal pieces are formed. Then what will be the ratio of surface area of the biggest piece to the smallest piece ?

Options:
A) 3.6
B) 3.2
C) 1.8
D) 2.4
E) None of these

The surface area of a sphere with radius, r, is 4Ï€r^2. As we make cuts, part of the surface area of the resulting pieces will be from a curved surface, and part will be from a flat surface. It may be helpful to keep track of these two sources of surface area separately. The surface area of the curved surfaces will be cut in half as we make each cut. The surface area of the flat parts will be made up of one or more portions of a circle with radius, r, that become exposed as a result of making the cuts.

1. Full sphere: Curved S.A.=4Ï€r^2 Flat S.A.=0
2. Hemisphere: Curved S.A.=2Ï€r^2 Flat S.A.=Ï€r^2
3. Quarter sphere: Curved S.A.=Ï€r^2 Flat S.A.=Ï€r^2 (two semi-circles with radius, r)

The next cut is defined somewhat ambiguously by the question. There are two ways that we could cut a quarter sphere "into a symmetrical fashion such that two equal pieces are formed". We could cut along the plane that runs through the center of the original sphere that is perpendicular to the semi-circular faces, or we could cut along the plane that runs from the edge created by the two semi-circles to the middle of the curved surface. Sort of like cutting an orange slice into two thinner orange slices. From the answer choices, I assume the former method is what was intended.

4. Half of quarter sphere: Curved S.A. = (Ï€/2)r^2 Flat S.A.=(3Ï€/4)r^2 (Because the exposed faces consist of three quarter circles of radius, r)

Now, the biggest piece is the hemisphere that we never cut, which would have a total surface area of 2Ï€r^2+Ï€r^2=3Ï€r^2, and the smallest piece is the piece produced in step 4, which has a total surface area of (Ï€/2)r^2+(3Ï€/4)r^2=(5Ï€/4)r^2.

So, the ratio of the biggest piece to the smallest piece is: 3/(5/4)=12/5= 2.4

Dear GMAT Expert,
Thanks for your reply.

I understood the derivation of curved surface area but for flat surface area, I am not able to understand the last part of (3Ï€r^2)/4.
How does this contain exposed faces of three quarter circles ?
When I imagine, then no matter into how many parts we cut a sphere, it will have only two flat surfaces. (Here I imagine the piece as a pastry-piece which a cut further from the quarter one.)

Plz help me understand this last part

In my solution, I made the cut this way:



Which leads to three exposed quarter circles: the one we can see, the one against the knife, and the one against the cutting board.

However, you could also make the cut like this:



in which case you would have only two exposed surfaces, both of them semi-circles.

Either interpretation is valid based on what is specified in the problem. I chose to do it the first way, though, because it gave me an answer that was actually one of the choices.

Simply superb !!!

I was actually imagining the 2nd way where two surfaces are exposed.

However, with the pictorial explanation you provided, I am through with the explanation

Thanks a ton !!!