Problem of the Day


Brooke uses a paper cutter to cut a 0.004 inch thick piece of paper in half. She then stacks the two halves on top of each other and cuts them in half again. She repeats the pattern until the stack becomes too thick for the paper cutter to cut. If the maximum thickness that the paper cutter can cut is 0.12 inches, how many pieces of paper will she have at the end of this process?

A. 16
B. 24
C. 32
D. 48
E. 64

Reveal Answer

Answer

C. 32

See the Solution

Solution

[latexpage]

Every time she cuts the paper and stacks the pieces, she is doubling the thickness of the previous stack.  Therefore, the thickness of the stack after $n$ cuts is $0.004 \cdot 2^n$ inches.  She is always able to make one more cut as long as the thickness of the stack does not exceed $0.12$ inches.  Symbolically, $0.004 \cdot 2^n \leq 0.12$.  Dividing both sides of this inequality by $0.004$ gives us $2^n \leq 30$, or $n \leq 4$.  This means that after 4 cuts, the thickness of the stack is still less than the maximum allowable thickness, $0.12$ inches.  So, she can do one more cut for a total of 5 cuts.  Every time she makes a cut, she doubles the number of pieces of paper, so after $n$ cuts, she will have $2^n$ pieces of paper.  So, after 5 cuts she will have $2^5=32$ pieces of paper.


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