Suppose that I cut a rectangular piece of paper in straight lines of equal length, parallel to one of the edges. If it took me s seconds to cut a paper into k pieces, how long will it take me to cut another paper into 4 k pieces at the same rate?
Details and Assumptions:
A cut is defined to be the line from one edge to another.
Papers are clean and fresh as new!
k ≥ 2 is a positive integer, and s is a positive real number.
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Can you explain what the first line means?
I think you want to say "we need to make k − 1 cuts".
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Argh. I probably rushed through the solution without checking thoroughly. Thanks for the note, Calvin!
I believe my explanation may not be perfect to some people's ears, so I believe there is some improvement for my solution. Hope it's the well-read solution. ^.^
Nice problem. It's impossible to say if you're allowed to fold the paper though :P
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That problem would be interesting to work on. If I can find the good answer and info for that one, I will post one.
I took, k = 2 , and k − 1 s ( 4 k − 1 ) was the only answer that gave 7 s , which I knew to be the answer for this extremely simple case. Since it was the only one that worked for the case it had to be the answer.
However, the correct solution is as follows:
k pieces requires k − 1 cuts. This took s seconds.
And for 4 k pieces you will require 4 k − 1 cuts.
So,
Time = cuts per second number of cuts = s k − 1 4 k − 1 = k − 1 ( 4 k − 1 ) s
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Since no cuts of the same length can cross each other, we can consider the following cases: 1 cut 2 cuts 3 cuts ( k − 1 ) cuts k cuts ⇒ 2 = (1 + 1) pieces ⇒ 3 = (2 + 1) pieces ⇒ 4 = (3 + 1) pieces ⋮ ⇒ k = ( ( k − 1 ) + 1 ) pieces ⇒ k + 1 pieces For each number of cuts, you increase that value by 1, which gives the number of pieces. From this relationship,
To make k pieces, we cut ( k − 1 ) times.
To make 4 k pieces, we cut ( 4 k − 1 ) times.
Then, k − 1 s = 4 k − 1 x = number of cuts time elapsed where x is the answer to the question. Thus, x = k − 1 s ( 4 k − 1 ) seconds for cutting into 4 k pieces.