Heartaches
1. Place all your hearts into a tray and count them. Remove any hearts that do nothave visible letters on one side.
2. Now shake your container vigorously but make sure not to lose or damage any hearts.
3. Remove the lid and then remove any hearts that have the letters up (you may eat these if you wish). Count the number of hearts remaining. (Note: If hearts land on their side, remove every other one of this type).
4. Continue this process until no hearts are remaining. Make sure to record your values in the following table:
Number of Shakes
Hearts
0
1
2
3
5. Now create two lists called SHAKE and HART on your calculator and record your data there. Do not enter the very last point (where hearts = 0).
6. Create a STAT PLOT of this data. What type of function would best model this data
7. Run an exponential regression on this data: y1 = r =
8. Remember, exponential functions are of the form y = ab. In your equation, a = and b = . Do these values make sense Explain:
9. Record the b values obtained by each of the other groups and yours: b = , , , , , , ,
10. Average these values. Apparently, b is very close to .
11. In Science, decay is often described with a life formula N = A(1/2 )^(t/h) where A is the original population, N is the final population, t is time, and h is the substance's life. Write the formula for N in our experiment if the half-life of these hearts is 1 shake. N =
12. How does this compare to the regression model
13. To the "average regression model"
14. Suppose the actual life of phenolbenzonate (what these yummy hearts are made of) is 13 years. If you currently have 60 grams of the substance, how much will you have in
a. 13 years
b. 26 years
c. 52 years
d. 100 years
15. Uranium-238 (U) decomposes into lead-206(Pb) with a half-life of 4.5 billion years. In a volcanic eruption, the lead is removed from the lava. If a sample of lava has 1 part PBto 99 parts U, how many years ago was the volcanic eruption that formed it.
16. Now, create a new list called LOG and set it equal to log(HART).
Create a STAT PLOT of SHAKE versus LOG. What type of regression would best fit this data
17. Run a linear regression on this data and place it into y2. y2 = r =
Compare this a and b to those in your exponential model. Is there a correlation?
18. From your exponential model: a1 = b1 =
19. From your linear model: a2 = b2 =
20. Now, find log a1 = and find log b1 =
21. How do these values compare to a2 and b2
22. In particular a2 = and b2 =
23. In general, any exponential curve of the form y = abcan be converted to the linear form y = mx + b2 by taking the logarithm of ab. Also, by logarithmic properties:
log(ab) = log a + log b
= log a + x log b
= (log b)x + log a
so that in the line y = mx + b2
m = the slope = and b2 = the y intercept =
24. In fact, by using logarithms, any exponential, power, or logarithmic function can be converted into a linear format. This is a valuable feature since lines are much easier to understand and analyze. Make use of previous notes dealing with regression models to answer the following:
a. (x, log y) is a line then (x, y) is an curve.
b. (log x, y) is a line, then (x, y) is an curve.
c. (log x, log y) is a line, then (x, y) is a curve.
25. Suppose I have the exponential equation y = 3 • 2.
a. What does 3 represent
b. Is this function increasing or decreasing
c. In linear format, y = mx + b where m = and b =