You have just landed your first summer job and your boss has told you that you can choose from one of 3 pay structures for the first week (7 days) that you work there. And after the first week, your pay will be frozen at the final rate for the remainder of the summer.
#1pay structure-1,3,5,7,9,11,13
#2pay structure-1,4,9,16,25,36,49
#3pay structure-1,2,4,8,16,32,64
1. Which option would you choose? Defend your decision mathematically.
2. Describe each option as linear, quadratic or exponential. Prove your description using finite differences.
3 obviously, 64 from now on is better than 49 or 13
1. linear, slope = 2
2. quadratic, 1^2 2^2 3^2 ..... 7^2
3. exponential 2^n
Thanks so much!
To determine which pay structure to choose, let's evaluate each option mathematically and analyze their growth patterns using finite differences.
1. Pay structure: 1, 3, 5, 7, 9, 11, 13
To find the pattern, let's calculate the differences between consecutive terms:
2 - 1 = 1
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
11 - 9 = 2
13 - 11 = 2
The differences between consecutive terms are constant, indicating a linear pattern. This pay structure increases by a fixed amount of 2 each day. The formula for calculating the pay on the nth day can be given as: Pay(n) = 2n - 1.
2. Pay structure: 1, 4, 9, 16, 25, 36, 49
Let's calculate the differences between consecutive terms:
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11
49 - 36 = 13
Again, the differences between consecutive terms are not constant. However, if we calculate the differences between these differences, we see that they are constant:
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
11 - 9 = 2
13 - 11 = 2
This indicates a quadratic pattern. The formula for calculating the pay on the nth day can be given as: Pay(n) = n^2.
3. Pay structure: 1, 2, 4, 8, 16, 32, 64
Calculating the differences between consecutive terms:
2 - 1 = 1
4 - 2 = 2
8 - 4 = 4
16 - 8 = 8
32 - 16 = 16
64 - 32 = 32
The differences between consecutive terms are not constant, and the differences between these differences are also not constant. This indicates an exponential pattern. The formula for calculating the pay on the nth day can be given as: Pay(n) = 2^(n-1).
Now, let's address the two questions:
1. Based on the three pay structures, mathematically, option #2 (Pay structure: 1, 4, 9, 16, 25, 36, 49) would be the best choice for maximizing pay. It follows a quadratic pattern, resulting in significantly higher wages compared to the other options. For example, on the seventh day, the pay would be 49 units (7^2) compared to 13 units for option #1 and 64 units (2^(7-1)) for option #3.
2. Option #1 is linear, option #2 is quadratic, and option #3 is exponential. We can determine this by analyzing the differences between consecutive terms. In option #1, the differences are constant, representing a linear pattern. In option #2, the differences are not constant, but the differences between differences are constant, indicating a quadratic pattern. In option #3, neither the differences nor the differences between differences are constant, suggesting an exponential pattern.