Problems

1. 1. We now know three ways to plot data: Cartesian (evenly spaced grid in both directions), semilog (logarithmic scale in one direction), log-log (logarithmic scale in both directions). In each case, seeing a straight line tells you something about the data. What does it tell you in each of these cases?
   2. What can you conclude about data if their plot is not straight in any of the three cases described in part a?
   3. Consider the points \((0,0)\), \((1,3)\), \((2,6)\), \((3,9)\), and \((4,12)\), all of which have \(y=3x\). On how many of the three types of plots would these data appear as a straight line? Which ones, and why?
   4. Consider the points \((0.1,-1)\), \((1,0)\), \((0,1)\), \((100,2)\), and \((1000,3)\), all of which satisfy \(y=\log_{10} x\). What kind of plot of these data would appear to be a straight line? Why?

2. Table P1   US population
   for the first 100 years

   Year	Population (millions)
   1790	3.929
   1800	5.308
   1810	7.240
   1820	9.368
   1830	12.866
   1840	17.069
   1850	23.192
   1860	31.443
   1870	38.558
   1880	50.156
   1890	62.948

    

    

   1. The data in Table P1 come from the early U.S. censuses. They have been entered in the Plot Data tool below, using "Census Number" as the unit of time. That is, "1" means the first census (1790), "2" the second (1800), and so on. Decide whether the U.S. population grew exponentially during its first century. If so, explain why you think so. If not, explain the way in which the data deviate from being exponential.

    

   2. Answer the same questions for the time from the first census to the start of the Civil War.

 

 

3. In our calculation of the linear relationship between \(\ln P\) and \(t\), we chose “ln” because the base of our exponential function was \(e\). Show that this choice was unnecessary. That is, given \(P=P_0 e^{kt}\), show that we could use logarithms with any base \(b\) and still find that “log of \(P\) is a linear function of \(t\).”
4. It may have occurred to you while working Activity 7 that you don't really need logarithms to tell whether the data in Table 3 fit \(s=c t^2\) — taking square roots would do as well.
   
   
   1. Verify this by calculating \(\sqrt{s}\) for the entries in Table 3 and testing for linearity as you did in that activity.
   2. Discuss the relative merits of these two procedures. Suppose you wanted to test whether \(s=c t^p\) for some unknown power \(p\). Could you do this by taking roots? Could you do it by taking logarithms? In the process, could you find out what \(p\) must be? Explain.
5. Use properties of exponentials and the definition of logarithms,

   \(y=\log_b x\)   if and only if   \(x=b^y\),

   to establish each of the following properties of logarithms.

   1. \(\log_b (A B )=\log_ A+\log_b B\)
   2. \(\log_b A^B=B \log_b A\)
   3. \(\log_b \frac{A}{B}=\log_b A-\log_b B\)
6. Use the property in Problem 5b to establish the "log conversion formula": For any two bases \(a\) and \(b\), and any positive number \(x\),
   
   
   \(\log_b x=\frac{\log_a x}{\log_a b}\).
   
   
   (Hint: Write \(y=\log_b x\) in exponential form, \(x=b^y\), take base-\(a\) logs of both sides, and substitute for \(y\).)
7. The formula in Problem 6 shows that every logarithm function is a constant multiple of every other logarithm function — specifically, the constant factor for converting \(\log_a x\) to \(\log_b x\) is \(1/\log_a b\). Write down explicit formulas for calculating \(\log_2 x\) by
   
   
   1. natural logarithms;
   2. common logarithms.
      
      
   Write the constant multiplier in each case with 4SD.
8. The user guide for a high-quality film camera usually includes a table of Exposure Values (EV's) that are determined by a combination of the shutter speed and the “f number” or aperture value, a measure of how wide open the shutter is when the film is exposed. The formula for calculating EV is
   
   
\(EV=\log_2 (f^2) -\log_2 (T)\),


where \(f\) is the f-number or aperture value, and \(T\) is the shutter speed in seconds. In practice, EV is rounded to the nearest integer.
1. Explain why — no calculation needed — the EV for an aperture value of 1 and a shutter speed of 1 second is 0.
2. Explain why each halving of the shutter speed (1/2, 1/4, etc.) increases the EV by 1.
3. Explain why each doubling of the f-value increases the EV by 2.
4. With one of your formulas from Exercise 13 and the pop-up calculator, find the EV for aperture value 8 and shutter speed 1/60 of a second. (Remember to round to an integer.)
5. Confirm your answer in part (d) by starting from \(f=1\) and \(T=1\) and successively doubling \(f\) and halving \(T\). You should be able to do this without a calculator. (Hint: In the world of cameras, when you halve 1/8, the answer is 1/15, so two more halvings gets you to 1/60.)


For more information about cameras and exposure values, including a complete EV table, visit T. Tamm's Exposure Value site.

9. Table P2   Spread
   of a Rumor

   Day	No. of Students Aware
   1	6
   2	11
   3	25
   4	45
   5	70
   6	130

   In Table P2, we show again the data from Activity 3. Suppose we model this data with a function of the form \(N=N_0 10^{mt}\).
   
   

   1. Show that \(m\) is the slope of the line in the semilog plot. Calculate this slope \(m\).
   2. Show that \(10^m=e^k\), where \(k\) is the slope of the \(\ln N\) plot. Conclude that \(10^{mt}=e^{kt}\) for all times \(t\), and that this model gives the same function as in Activity 3.
10. Show that if the data represent a function of the form \(P=P_0 e^{kt}\), then \(\log_b P\) will be a linear function of \(t\) no matter what the base \(b\) is.
11. In Table P3 we repeat the data from our power model Example. Show that it would not have made a difference in this Example if we had used natural logarithms instead of common logarithms.
    
    

    Table P3   Toy dinosaur stuffing

    x	1	2	3	4	5	6	7
    v	2.4	19.1	64.0	152	295	510	813

12. 1. In Table P4 we show the number of miles of railroad track in the United States in the late nineteenth century. Determine the growth pattern, and find a formula for a function that fits these data.
       
       

    Table P4 Miles of railroad track in the United States

    Year	1860	1870	1880	1890
    Miles of Track	30,626	52,922	93,262	166,703

    
    
    2. If that pattern you found in part (a) had continued for another century, how many miles of track would there have been in the continental United States in 1990? What percentage of the area of the United States would have been covered by railroad beds in 1990?
    3. In what year would this growth pattern have led to covering the entire country with railroad bed?
    4. Suppose your estimate of the area of the United States is off by 10% one way or the other. How would that affect your answer to part c?
    5. Suppose your estimate of the width of a roadbed is off by a factor of 2 one way or the other. How would that affect your answer to part c?
    
    
    Problem suggested by R. H. Romer, “Covering the U.S.A.: Exponential Growth of Railroad Tracks,” The Physics Teacher 28 (1990), pp. 46–47, and A. A. Bartlett, “A World Full of Oil,” The Physics Teacher 28 (1990), pp. 540–541.