In our last post, we dove into one of the hardest math problems I have seen on an official ACT practice test.  We used some helpful strategies to narrow us down to two options, even without knowing how the problem works.

Today, we talk more about the actual math concepts related to this problem, and how we can apply these to get the correct answer.

So alright, the question is asking about arc length.  Most of you saw questions like this towards the end of the year in geometry, so rack your brains and try to remember what your teacher was going over while you were daydreaming about summer vacation.  There’s a proportion that relates arc lengths, the circumference of the circle, and the angle making that arc.

                 Arc length                           angle

                 ———————       =    ———————

            Circumference                           360

 

Look familiar?  Alright, well maybe not, but this is a helpful proportion to know for the ACT and SAT (there is a similar equation relating area of a sector in a circle to that circle’s area).  Additionally, this should make some intuitive sense.  After all, an arc is really just a fraction of the circle’s circumference.  And the angle making that arc is just a fraction of the 360 degrees found in every circle.  Makes sense that those fractions would equal each other.  

 

Almost there.  We know the question is asking us for arc length, and we have an equation that can be used to find arc length, so let’s plug in what we know.

 

The circumference of a circle is 2𝞹r (NOT 𝞹r^2 as countless students have told me in the past) so for this circle we’ve got 2𝞹(4) or 8𝞹.

 

       Arc length                                angle

            ———————-       =          ———————-

             8𝞹                                         360

We can get arc length by itself by multiplying both sides by 8𝞹 (or by cross-multiplying and dividing by 360), and in either case we get

 

                Arc length   =                 8𝞹(angle)

                                                   ———————–

                                                     360

But what is the angle, you might be asking?  Well, that’s what we figured out last time!  Because we know the opposite side of the angle is 1 and the hypotenuse is 4, the sine of our angle is (¼) and the angle can then be written as sin-1(¼).

 

                                                   8𝞹(sin-1(¼))

                Arc length   =          —————-

                                                   360

 

Looking very close now.  Maybe we can reduce this fraction?  8𝞹 and 360 are both divisible by 8, so we can reduce this we get

                                                    1𝞹(sin-1(¼))

                Arc length   =          ——————

                                                    45

Or answer choice A.

Phew.  Well if nothing else, I hopefully have at least convinced you that this question might just be the hardest the ACT has to offer.  As I said last time, however, there are some very important lessons that can be taken from approaching a question like this, so let’s briefly recap those.

  1. If you find yourself totally mystified by a question on the ACT (particularly in the math section) ask yourself what type of math concept is this testing?
  2. A good source of information/inspiration when thinking about a question is the answer choices!  We saw sine, cosine, and tangent in all the choices, which gave us a major clue about how to narrow down our choices.
  3. If you can think of any equations or helpful rules related to question, write them down!  Here we utilized SOH CAH TOA and the arc length proportions, and having them written down in front me me is extremely helpful when I’m trying to figure out what to do.

A piece of encouragement to end on: the strategies, techniques, and concepts that we used to solve this problem are absolutely learnable, as is everything else on the ACT and SAT.  Plus, after this question, everything else will be a walk in the park!