<p>If anyone could do this problem and send me a picture of how to do it, I would greatly appreciate it. The problem is 2 circular pulleys with centers 8 inches apart are connected with a tight belt. The belt wraps 2/3 of the way around the larger puller, which has a radius of 5 inches, and 1/3 of the way around the smaller pulley, which has a radius of 1 inch. What is the exact length of the belt, in inches? This question was from the April 2014 ACT and I can't quite figure out how to do it. The answer is 22pi/3+8squareroot(3). I have no problem getting the 22pi/3 but coming up with the 8squareroot(3) is an issue. Thanks to anyone who helps.</p>
<p>I’ll try to explain this as clearly as I can w/out a picture…</p>
<p>Let O and P denote the centers of the smaller and larger pulleys respectively. Draw radii OA and PB such that OA and PB are parallel, and both perpendicular to the straight section of the pulley (AB). Let D be on PB such that OD is parallel to AB. Then OPD is a right triangle, and you can determine that OP = 8, PD = 4. Then OPD is a 30-60-90 triangle, and OD = AB = 4 sqrt(3). Since that is the length of 1/2 of the straight part, the length of the straight part is 8 sqrt(3).</p>
Bump.
@MITer94 I’m still confused about this problem. How do you have PB such that OD is parallel to AB? o_o
@sdw8253 try drawing it out, and read my explanation carefully. Note that POAB is a trapezoid, where AB is half of the “straight” part of the pulley (which is what you want to figure out).
You don’t need to do any Math. That’s a waste of time. Just estimate.
8 square root 3 is approx equal to 14. That represents the total length of the belts extending from the small circle to the large circle. Each one is approx 7.
If you look at the line marked “8”, then you know that the belts are about the same length as that line marked “8”.
The other answers are either too small or too big.
@mmk2015 estimation sometimes works, and it may be a good strategy for certain people or for specific problems, but by estimating it’s often difficult to ascertain whether the answer is correct. You said that the two belt lengths have length roughly 8, but in fact their length must be less than 8.
Note that this problem was solved using very little math - only some drawing/constructing segments and noticing a 30-60-90 triangle. The hard part was explaining the solution without a diagram.
That’s why I said “approx”. But you can still get the answer this way.
To go one step further, you can approximate that the length of the entire belt wrapped around the pulleys is 5 of those lines of length “8”, which is a total of 40.
If you use a calculator to convert the answers into numbers, the answers look like this:
A) 31
B) 37
C) 61
D) 65
E) 67
The answer is clearly B, without any complicated Math.
You can either spend 5 mins trying to do real Math and prob still get it wrong, or spend 1 min appoximating, checking the answers and prob get it right.
Also, if you’re struggling to explain the solution without a diagram, then the problem is not as easy as you imply.
For 90% of the country, this problem is pretty hard using straight Math. But I’m sure most can approximate.
@mmk2014 I never said approximating was a bad or wrong solution, but for me at least, on many cases it’s harder to verify if the answer is correct. If it works for you, great.
However approximating won’t really work for any non-multiple choice questions (such as SAT grid-ins), or ACT questions where two or more choices are really close together. For me it wasn’t immediately obvious that 31 was out, but C,D,E clearly were.
On some questions though, I’ll agree that it is much faster to approximate. For example:
[Set 4 Question 8](http://www.actstudent.org/sampletest/math/math_04.html)
The area of the square is 10^2 = 100, and the area of each semicircle is a little less than 50 (alternatively, the area of each full circle is less than 100, but greater than 50). Only choice H makes sense.
ACT Quantumm has a video on this problem if that helps.