full-length 4, ps #35

I just don't understand number 35 at all, could you explain it? Thanks!
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For this question, it asks us to find the length of the “sweet zone” on a 81 cm bat.  Let’s first consult our passage for the definition of “sweet zone”:

“The sweet spot is the node of the second harmonic nearest the thicker end of the bat…there is a ‘sweet zone’ that extends from the old sweet spot to the node of the third harmonic which is closest to the thick end of the bat.”

Alright, so we’re dealing with standing waves (based on the words “node” and “harmonic”).  Earlier in the passage, it informs us that the bat resonates much like an open pipe.  Let’s take a look at a diagram of the second and third harmonics (n = 2 and n = 3).  Remember that the a harmonic for an “open-at-both-ends” or “closed-at-both-ends” tube is the number of half-wavelengths.

n = 2 

n = 3 

Considering that our bat is 81 cm long, let’s locate the node in the second harmonic. Each node is ¼ the distance of the bat from each end; divide the bat into four equal lengths and you’ll see that you have either a node or antinode at each division.  So the node for the n = 2 bat is 81/4 = 20.25 cm from the end.

For the third harmonic, let’s do a similar process.  The node closest to the end is 1/6 the length from the end of the bat – again, look at the third harmonic diagram above, and you’ll see that dividing the bat into six equal lengths will give you a node or antinode at each end.  The node closest to the end is 1/6 of the way over, which is 81/6 = 13.5 cm.

So the distance between them is 20.25 cm – 13.5 cm = 6.75 cm.  This, thus, represents the length of the “sweet zone”.