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Population Aging The following table shows the percentage of U.S. residents over the age of 65 in 1950, 1960, …, 2010:38
Year t (years since 1900) | 50 | 60 | 70 | 80 | 90 | 100 | 110 |
Percentage P over 65 (%) | 8.2 | 9.2 | 9.9 | 11.3 | 12.6 | 12.6 | 13 |
a. Find the logarithmic regression model of the form
b. In 1940, 6.9% of the population was over age 65. To how many significant digits does the model reflect this figure?
c. Which of the following is correct? The model, if extrapolated into the indefinite future, predicts that
(A) The percentage of U.S. residents over the age of 65 will increase without bound.
(B) The percentage of U.S. residents over the age of 65 will level off at around 14.2%.
(C) The percentage of U.S. residents over the age of 65 will eventually decrease.
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Chapter 2 Solutions
Applied Calculus
- Cable TV The following table shows the number C. in millions, of basic subscribers to cable TV in the indicated year These data are from the Statistical Abstract of the United States. Year 1975 1980 1985 1990 1995 2000 C 9.8 17.5 35.4 50.5 60.6 60.6 a. Use regression to find a logistic model for these data. b. By what annual percentage would you expect the number of cable subscribers to grow in the absence of limiting factors? c. The estimated number of subscribers in 2005 was 65.3million. What light does this shed on the model you found in part a?arrow_forwardSales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. Table 4 shows the number of games sold, in thousands, from the years 20002010. a. Let x represent time in years starting with x=1 for the year 2000. Let y represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data. b. If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest thousand.arrow_forwardWorld Population The following table shows world population N, in billions, in the given year. Year 1950 1960 1970 1980 1990 2000 2010 N 2.56 3.04 3.71 4.45 5.29 6.09 6.85 a. Use regression to find a logistic model for world population. b. What r value do these data yield for humans on planet Earth? c. According to the logistic model using these data, what is the carrying capacity of planet Earth for humans? d. According to this model, when will world population reach 90 of carrying capacity? Round to the nearest year. Note: This represents a rather naive analysis of world population.arrow_forward
- What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?arrow_forwardTable 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012. a. Let x represent time in years starting with x=0 for the year 1997. Let y represent the number of seals in thousands. Use logistic regression to fit a model to these data. b. Use the model to predict the seal population for the year 2020. c. To the nearest whole number, what is the limiting value of this model?arrow_forwardTorontos Jewish Population The table gives the population of Torontos Jewish community at various times. Source: The Mathematics Teacher. Plot the population on the y-axis against the year on the x-axis. Let x represents the years since 1900. Do the data appear to lie along a straight line? Plot the natural logarithm of the population against the year. Does the graph appear to be more linear than the graph in part a? Find an equation for the least squares line for the data plotted in part b. If your graphing calculator has an exponential regression feature, find the exponential function that best fits the given data according to the least squares method. Take the natural logarithm of the equation found in part d, and verify that the result is same as the equation found in part c. In Section 11.1 on Solutions of Elementary and Separable Differential Equations, we will see another type of function that is a better fit to these data.arrow_forward
- EXERCISES The following table gives the life expectancy at birth of females born in the United States in various years from 1970 to 2010. Source: National Center for Health Statistics. Year of Birth Life Expectancy years 1970 74.7 1975 76.6 1980 77.4 1985 78.2 1990 78.8 1995 78.9 2000 79.3 2005 79.9 2010 81.0 Find the life expectancy predicted by your regression equation for each year in the table, and subtract it from the actual value in the second column. This gives you a table of residuals. Plot your residuals as points on a graph.arrow_forwardWhat situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.arrow_forwardLong-Term Data and the Carrying Capacity This is a continuation of Exercise 13. Ideally, logistic data grow toward the carrying capacity but never go beyond this limiting value. The following table shows additional data on paramecium cells. t 12 13 14 15 16 17 18 19 20 N 610 513 593 557 560 522 565 517 500 a. Add these data to the graph in part b of Exercise 13. b. Comment on the relationship of the data to the carrying capacity. Paramecium Cells The following table is adapted from a paramecium culture experiment conducted by Cause in 1934. The data show the paramecium population N as a function of time t in days. T 2 3 5 6 8 9 10 11 N 14 34 94 189 330 416 507 580 a. Use regression to find a logistic model for this population. b. Make a graph of the model you found in part a. c. According to the model you made in part a, when would the population reach 450?arrow_forward
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