Consider the following problem. Mike and Tim are looking to earn a little extra
ID: 2912739 • Letter: C
Question
Consider the following problem. Mike and Tim are looking to earn a little extra money. The beach committee offers them the opportunity off picking up plastic bottles, paying them $0.20 per bottle. Mike realizes as they pick up it will get harder and harder to find more bottles. So as an incentive to keep looking he suggest a different form of payment. He suggests $0.10 for the first bottle, and increase the pay by 2% for each bottle after that. Tim thinks Mike is crazy to propose an increase of just 2% per piece. He plans to ask for a one-cent increase for every piece, starting at 15 cents for the first bottle. Suppose the committee accepts each offer. x Answer the following in detail: For Mike and Tim, write the first 5 terms of a sequence for payment scheme for each piece of paper. Write a function to generate the n-th sequence values (bottle payment) for each Mike and Tim. Explain why you chose that function. Using the formula how much will each receive for the 50th bottle? For the 100th bottle? Determine the type of sequences for both Mike and Tim. Who do you think fairs better in the long term? Explain.
Explanation / Answer
Mike :
0.10 then an increase of 2%
Tim :
0.15 and then an increase of 0.01
Mike :
0.1 , 0.1*1.02 , 0.1*(1.02)^2 , 0.1*(1.02)^3 , 0.1*(1.02)^4
0.1 , 0.102 , 0.10404 , 0.1061208 , 0.108243216
Tim :
0.15 , 0.16 , 0.17 , 0.18 , 0.19
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Write a function to generate the n-th sequence values (bottle payment) for each Mike and Tim.
Mike :
Every nth bottle gets 0.10 * (1.02)^(n-1) --> ANS
Tim :
First bottle got him 0.15
Then 0.01 gets added etc
So, using sum of an arithmetic series,
0.15 + (n - 1)(0.01)
0.14 + 0.01n ---> ANS
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Explain why you chose that function.
Well, for Mike, it's an exponentially increasing one.
So, 1.02 is 1 + 2% increase written as a decimal.
With each subsequent bottle, he gets 1.02* previous amount
So, we chose an exponential
Tim :
Clearly for Tim, the function is linear because he wants an increment of 1 cent(0.01 dollars) after each bottle pick.
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Using the formula how much will each receive for the 50th bottle? For the 100th bottle?
50th bottle :
Mike = 0.10 * (1.02)^(n-1)
0.10 * (1.02)^49
$0.26
Tim : 0.14 + 0.01n
0.14 + 0.01(50)
0.64$
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100th bottle :
Mike : 0.10 * (1.02)^99 ---> $0.71
Tim : $1.14
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Mike will perform better in the ,ong run because exponential while starting slow will zoom forward later compared to linear.