You must divide a rectangle into 3 equal parts along it"s length. But you have the right to only division things into half. Friend don"t have any type of other tools available - a saw/cutter and also a scale that have the right to accurately tell you the fifty percent of a complete length. How can this be done?
Assume a piece of wood:
Length: (16x) ----------------Step 1: (2 halves) --------|-------- step 2: (4 quarts) ----|----|----|---- (1/3rd is somewhere in between the very first and 2nd cut)And so on till you space close to $1/3$ with some $\pm\epsilon$. How can I ptcouncil.netematically frame and also solve this problem and say after how numerous divisions will certainly I it is in close come the 1/3 mark?
edited may 30 "13 in ~ 18:07
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If we aren"t pertained to with exactitude and also are ready to accept part $\pm\epsilon$ then it is just a question of "how much" error we deserve to tolerate.
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One can mark down halves successively to acquire lengths that $\frac12^n$
Once you acquire that measurement as an agree unit, you have the right to count $x$ systems off to acquire to $\frac13$ and also $\frac23$
Let me to walk you with a straightforward example:
Say the length of the block is 100.
Then we want to reduced it turn off at 33.33 and also 66.67.
Now successively measure up halves (half of fifty percent of fifty percent of...) 4 times. This gives us a unit the $\frac116*100 = 6.25$
Now we can count off 5 systems as 31.25, and also 6 systems as 75, which room "pretty close" come 33.33 and also 66.67 respectively.
If your error yongin is low, you have the right to just save making the unit smaller till friend fit inside your $\pm\epsilon$ range.
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In the above example, if you had successively measure halves 8 times, rather of 4, the unit would have actually been $\frac1256*100 = 0.390625$
Counting off 85 systems = 33.20 (yup, yes, really close to 33.33)Counting turn off 171 units = 66.79 (reasonably close come 67.67)In truth you can do one better. ~ counting off 85 units and also cutting that piece off, you are left v 66.80. Now just chop that off right into half, and you"ll end up through 3 pieces: 33.20, 33.40, 33.40
The benefit of utilizing the size as 100 is this: every the numbers above can actually be expressed as %.
Now if we had to issue a ceiling statement, regarding how countless successive steps of half it take away to come at the ideal unit, you might say$max(\frac12^n l)