As shown in the figure above, a thin conveyor belt 15 feet long is drawn tightly around two circular wheels each 1 foot in diameter. What is the distance, in feet, between the centers of the two wheels.

I don't understand how to set up this problem. Every approach I take doesn't really answer the question. I estimated the answer to be 5.5, which leads me to the correct answer, but how do I solve it mathematically?

To set up the problem, the key takeaway is that 15ft is the entire length of the oval around the two circular wheels.

Step 1: bisect each of the circles with a vertical diameter running through the center. The half circles at the ends are the portion of the belt wrapped around the circular wheels.

Step 2: figure out the circumference of the two wheels.
C = 2*pi*r = 2*pi*(1/2) = pi

Step 3: add the conveyor belt lengths of the left and right circles together (these are the half circles)
(1/2)*pi + (1/2)*pi = pi

Step 4: Thus, 15-pi is equal to the length of the two parallel lines between each of the two centers.

Step 5: Divide by two to get the distance between the centers of the two wheels
15-pi
2

mxs2009's solution is extremely well executed.

like so many other geometry problems, this one is also a problem on which ESTIMATION can go a long long long way.

the whole conveyor belt is 15 feet long. even if you just folded it in half, with no circular parts, each half would only be 7.5 feet long. therefore, it's clear that the desired distance is less than 7.5 feet.

in any case, if you "guess-timate" the distance, you should come up with something in the range of 4 to 6 or so.

(c) and (d) are absurd; both of them are greater than 7.5 feet. (you can figure this out quickly with the primitive assumption that pi is about 3)

(b) is really really small. 5/4 times 3 is only about 3.75.

(a) and (e) are both approximately 6 (using the same estimate that pi is about 3).
however, if you have to guess between these two, you should be loath to pick (e), since
* its form is SO DIFFERENT from that of the other four answers, and
* there is nothing in it that is at all related to the "15" (a quantity that clearly matters).

that leaves (a).

estimation for the win.