Sample Exercises  
 Chapter 7: Solving Equations and Fitting to Data
 
 
  1. In 1980 there were approximately 1,200,000 African elephants.  In 1990 there were approximately 700,000 African elephants.  Assume that the population growth factor is constant.

    2.  
    1. Find the function that models these data where the input is the time since 1980 and the output is the number of African elephants.
    2. Use your function to estimate what the size of the population of Africa elephants will be in 2010.
    3. Use your function to estimate the year when the African elephant population will be half the size it was in 1980.
    4. When will the African elephant population be half the size it was in 1990?
     
  2. Moore's Law, created by Intel co-founder Gordon Moore, states that microprocessors in computers will get twice as powerful for the same price every 18 months.
    1. If Moore's Law is true, what is the annual growth rate in the power of micro processors?
    2. If Moore's Law is true, what is the continuous annual growth rate in the power of micro processors?
     
  1. Based on a Bureau of Justice Statistics report, a newspaper headline read, ``U.S. prison population doubled in 12 years."
    1. Assuming exponential growth, what was the annual growth rate for the prison population during those 12 years?
    2. If the prison population continues to grow exponentially at the same rate, how long will it take the population to triple?  To quadruple?
     
  1. Life expectancy at birth has increased during the last hundred years. Since 1920, female life expectancy has been greater than male life expectancy. This can be seen in the following table.
     
    year 1920 1930 1940 1950 1960 1970 1980 1990
    male life expectancy 53.6 58.1 60.8 65.6 66.6 67.1 70.0 71.8
    female life expectancy 54.6 61.6 65.2 71.1 73.1 74.7 77.5 78.8
     
    1. Make a scatterplot of the male life expectancy on the horizontal axis and the female life expectancy on the vertical axis. Do these data fall in a linear pattern?
    2. Find the regression equation where the input is male life expectancy and the output is female life expectancy.
    3. Based on your regression equation, is the gap between male life expectancy and female life expectancy increasing or decreasing. How can you tell this just from the equation?
     
  1. While CD sales increased greatly in the late 1980’s and early 1990’s, record sales decreased greatly. The following table gives the number of single record sales in millions.
     
    year 1985 1987 1988 1989 1990 1991 1992 1993 1994
    single record sales 
    (in millions)    		 120.7 82.0 65.6 36.6 27.6 22.0 19.8 15.1 11.7
     
    1. Find the logarithm of the record sales for each year given.
    2. Find the linear regression equation where the time since 1980 is the input and the logarithm of the record sales is the output.
    3. Transform your linear equation into an exponential equation where time since 1980 is the input and the record sales is the output.
    4. How well does your exponential equation fit the data?
     
  1. Two hundred thumbtacks were tossed onto a table. The ones that landed point up were removed. The remaining were again tossed onto the table and again the ones landing point up were removed. This process continued until all of the thumbtacks were removed from the table. The results are shown in the following table.
 
tosses 0 1 2 3 4 5 6 7 8
number of tacks remaining 200 76 28 12 10 1 1 1 0
 
    1. Why does it make sense that an exponential function would fit a situation like this?
    2. Why will the last point, (8,0), make it impossible to use exponential regression on the data?
    3. Eliminate the last point and find an exponential equation where the input is the toss number and the output is the number of thumbtacks remaining.
    4. How would you think your exponential function would change if instead of removing the thumbtacks that landed point up, the ones not landing point up were removed?
  1. In a book about the biology of birds, the equation log M = log 89 + 0.64 log W is given. In this equation, W is the weight of a bird in kilograms and M is the bird's metabolic rate in kilocalories per day.  The given equation can be considered a linear equation where the input is log W and the output is log M.
    1. Transform this equation into one where the input is W and the output is M. It is stated in the book that a twofold increase in the body weight of a bird is accompanied by less than a doubling of the metabolic rate.
    2. Using your transformed equation from part (a), explain why this is true.
  1. The following table gives the average weight and length for babies according to the National Center for Health Statistics.
 
Age (in months) 3 6 9 12 15 18 21 24
Weight (in pounds) 13.0 17.2 20.3 22.2 24.0 25.3 26.6 27.8
Length (in inches) 24.0 26.7 28.6 30.0 31.4 32.5 33.6 34.5
 
    1. Make a graph of both age versus weight and age versus length. (Both of these should have age on the horizontal axis.) Explain why power functions should do a better job than exponential functions of fitting each of these plots.
    2. Find a power function that fits the data where age is the input and weight is the output.
    3. Find a power function that fits the data where age is the input and length is the output.
    4. Combine your two power equations in such a way that length is the input and weight is the output. Is this function also a power function?
     

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