## Help with Exponential Growth...

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
cori1984
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### Help with Exponential Growth...

A cleaning service will clean your house for the month of December. You will pay the service 1 cent at the beginning of December 1. Let x represent the number of complete days worked. On December 1 the number of complete days worked is zero and x=0, on December 2 one complete day has been worked and x=1, on December 3 two complete days have been worked and x=2 . Y represents the amount paid. On December 1 the amount paid is 1 cent, on December 2 the amount paid is 2 cents, on December 3 the amount paid is 4 cents, on December 4 the amount paid is 8 cents.

Make a table of independent variables (x=complete days worked) and dependent variables (y=amount paid on that date) through t=20.
Choose an appropriate scale for the points to be plotted.
Plot the points using the TI calculator. Copy and paste this scatter plot into your paper using the TI connect. If necessary you can use Winplot. Define a model of uninhibited growth A = A0ekt mathematically.

Use the calculator to define a model of uninhibited growth in the form y = abx . Show that if A=y and t=x then the two functions y = abx and A = A0ekt are the same.

Describe where the graph is increasing, decreasing, or constant. Define the Domain and the Range of the function. How much will you be making on the 15th day? The 20th day?
How much did the cleaning service total over the first 20 days?

stapel_eliz
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You've been provided with fairly detailed instructions. At what point are you getting stuck?

cori1984
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Joined: Thu Nov 19, 2009 8:08 pm
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### Re: Help with Exponential Growth...

I kind of not sure about anything. I have been struggling with this section and just need the foundation to get started. I thought I had the table right, I believed it was doubling each day but as I got higher up I started getting amounts way higher than what I would pay for 20 days of house cleaning. I guess my first question would be what should the table look like?

stapel_eliz
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On what basis did you assume that this math question was applicable to the "real world"? The huge values are largely to point of this exercise!

So work with the numbers, make the table, fill in the values for the formula, and plug stuff into the software they've given you. Then, besides having a good grasp of the meaning of "exponential growth", you'll also be prepared if anybody ever runs the chessboard problem past you!

cori1984
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### Re: Help with Exponential Growth...

Thank you for that information. When I just double everything I get 5,242.88 for the 20th day. But if I follow the equation given in the wheat and chessboard example, T_20=2^20-1, I get 1,048,575. Which one is correct? Should the 20th day be 5,242.88 or 10,485.76? Below is the table I created
1. .01
2. .02
3. .04
4. .08
5. .16
6. .32
7. .64
8. 1.28
9. 2.56
10. 5.12
11. 10.24
12. 20.48
13. 40.96
14. 81.92
15. 163.84
16. 327.68
17. 655.36
18. 1310.72
19. 2621.44
20. 5242.88

I believe this would be correct so the equation should really be 2^19-1 correct? So the graph would be constant since it increases at the same rate? The range would be [.01,5242.88]? The domain would be [1,20]? The person would be making 163.84 on the 15th day and 5,242.88 on the 20th day? The cleaning service will total 10,485.75 over the first 20 days? How do I create the equations requested? (A=A_0e^kt and y=abx) The calculator said for the y=ab^x equation a=0.005 and b=2. So that equation would be y=(.005)(2)^x right? That would mean the A=A_0e^kt would be .01e^(.005)(t)? or in order to make it equal to y=(.005)(2)^x would it be .005e^(.6931)(t)? Thanks again for the help.

stapel_eliz
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Joined: Mon Dec 08, 2008 4:22 pm
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I don't know what you mean by "constant", since that usually refers to linear growth, which this clearly isn't.

As for the formula, take a closer look at it. For 64 squares, there was the first square, followed by 63 doublings. What is the power on the 2? 64? Or 63?

As for the domain and range, I'm not sure if they're referring to the whole-number domain (from 1 to 20) for the word problem, or the "all real numbers" domain for the function. They say "function", but you might want to ask your instructor, just to be sure...?