An architect is planning several stone spheres...

[ZacharyS701]

An architect is planning several stone spheres of different sizes into the landscaping of a public park, and workers who will be applying the finish to the exterior of the spheres need to know the surface area of each sphere. The finishing process costs $92 per square meter. The surface area of a sphere is equal to 4(pi)r2 where r is the radius of the sphere.
How much would it cost to finish a sphere with a 5.50-meter circumference and a 7.85-meter circumference?

The answer options are:
$900
$1200
$1800
$2800
$3200
$4500

The answers are $900 for the 5.5 meter, and $1,800 for the 7.85 meter

This makes no sense to me. Any help would be appreciated.

[ZacharyS701]

Initially, I tried to isolate the value of r. Circumference=2(pi)r, therefore one could isolate r. This gave me numbers that were very inflated according to the actual answers. I got $1,800 for the 5.5 meter, and over $4,500 for the 7.85-meter sphere.
I then realized the following relationship: The surface area of a sphere is 4(pi)r2 and circumference is 2(pi)r. Therefore, the surface area of a sphere is merely the circumference times 2r. I tried to use this reasoning, but the calculations don't add up because the $900, and $1,800 answers are not for the surface area of the sphere, but the surface area times $92 per square meter finishing cost.
I am completely lost on this on. Any help would be appreciated.

[Sage Pearce-Higgins]

Your logic is good, let me try to see where you're going wrong. If you know that the circumference is 5.5m, then you can work out the radius using the formula C = 2 (pi) r. This gives a radius of approx 0.88 meters. Then use that to work out the surface area, approx 9.68 square meters, then multiply that by 92 which gives around $900. Try this out for the other sphere. Note also that you have a calculator and you only need to be approximate with your answer.

Initially, I tried to isolate the value of r. Circumference=2(pi)r, therefore one could isolate r. This gave me numbers that were very inflated according to the actual answers. I got $1,800 for the 5.5 meter, and over $4,500 for the 7.85-meter sphere.

Notice that these aren't just inflated, they're about double what you would hope for. You probably forgot to divide by 2 at some stage.

I then realized the following relationship: The surface area of a sphere is 4(pi)r2 and circumference is 2(pi)r. Therefore, the surface area of a sphere is merely the circumference times 2r. I tried to use this reasoning, but the calculations don't add up because the $900, and $1,800 answers are not for the surface area of the sphere, but the surface area times $92 per square meter finishing cost.

This is good - there's a shortcut here. However, how can you multiply the circumference by 2r if you haven't worked out r? Answer, if the circumference is 2(pi)r, then the surface area is (circumference^2 ) / pi. This works out to give the right answer.

[KarlS319]

Thanks for posting this IR question. Followed the correct logic when attempting the question in a practice exam setting but must have messed up somewhere along the lines while punching #s into the calculator. To add to the explanation, you are basically "undoing" or working backwards from the given circumference. The exact steps to solve for each circumference would be as follows:

(1) divide by pie to get your diameter
(2) divide by two to get your radius
(3) calculate the surface area of the sphere according to the given formula (four)(pie)(rsquared)
- square your radius (BTW is there a way to quickly do this using the GMAT calculator or do you have to manually times the number by itself?)
- times by pie and then times by four.
(4) Now that you have your surface area multiply by the cost $92 per square meter to get the total finishing cost

In each case, your numbers should come out roughly around $900 for the 5.50m circumference and $1,800 for the 7.85m circumference.

[Sage Pearce-Higgins]

That seems to work. In answer to your question, the calculator has just simple functions, so there's no way to square a number without retyping it. However, given that these answers are spaced relatively far apart, you don't need to be too precise with the digits.