Concept explainers
1–16 ■ Population Growth These exercises use the population growth model.
Bat Population The bat population in a certain Midwestern county was 350, 000 in 2012, and the observed doubling time for the population is 25 years.
(a) Find an exponential model
(b) Find an exponential model
(c) Sketch a graph of the population at time t.
(d) Estimate how long it takes the population to reach 2 million.
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Algebra and Trigonometry (MindTap Course List)
- Population The table shows the mid-year populations (in millions) of five countries in 2015 and the projected populations (in millions) for the year 2025. (a) Find the exponential growth or decay model y=aebt or y=aebt for the population of each country by letting t=15 correspond to 2015. Use the model to predict the population of each country in 2035. (b) You can see that the populations of the United States and the United Kingdom are growing at different rates. What constant in the equation y=aebt gives the growth rate? Discuss the relationship between the different growth rates and the magnitude of the constant.arrow_forwardPopulation Growth These exercises use the population growth model. 10. Bat Population The bat Population in a certain Midwestern county was 350,000in 2012. and the observed doubling tune for the is 25 years. (a) Find anexponential model n(t)=n0et/a for the Populationt years after 2012. (b) Find an exponential model n(t)=n0ert the Population t years after 2012. (c) Sketch a graph of the population at time t. (d)Estimate long it takes to reach 2 million.arrow_forwardPopulation Growth These exercises use the population growth model. 1. Bacteria Culture A certain culture Of the bacterium Streptococcus A initially 10 and is to every 1.5 hours. (a) Find an exponential for the number of bacteria in culture after t hours. (b) Estimate the number of bacteria after 35 hours. (c) After how many hours will the bacteria count reach 10,000?arrow_forward
- U.S. Population The U.S. Constitution requires a census every 10 years. The census data for 17902010 are given in the table. a Make a scatter plot of the data. b Use a calculator to find an exponential model for the data. c Use your model to predict the population at the 2020 census. d Use your model to estimate the population in 1965. Year Population in millions Year Population in millions Year Population in millions 1790 1800 1810 1820 1830 1840 1850 1860 3.9 5.3 7.2 9.6 12.9 17.1 23.2 31.4 1870 1880 1890 1900 1910 1920 1930 1940 38.6 50.2 63.0 76.2 92.2 106.0 123.2 132.2 1950 1960 1970 1980 1990 2000 2010 151.3 179.3 203.3 226.5 248.7 281.4 308.7arrow_forwardPopulation Growth These exercises use the population growth model. 2. Bacteria Culture A certain culture or the bacterium Rhodo-bacter sphaeroides initially has 25 bacteria and is to double every 5 hours. (a) Find an exponential model n(t)=n02t/a for the number of bacteria in the culture after t hours. (b) Estimate the number Of bacteria after 18 hours. (c) After how Will bacteria reach 1 million?arrow_forward116 Population Growth These exercises use the population growth model. Deer Population The graph shows the deer population in a Pennsylvania county between 2010 and 2014. Assume that the population grows exponentially. a What was the deer population in 2010? b Find a function that models the deer population t years after 2010. c What is the projected deer population in 2018? d Estimate how long it takes the population to reach 100, 000.arrow_forward
- Growth Rate Versus Weight Ecologists have studied how a populations intrinsic exponential growth rate r is related to the body weight W for herbivorous mammals. In table 5.2, W is the adult weight measured in pounds, and r is growth rate per year. Animal Weight W r Short-tailed vole 0.07 4.56 Norway rat 0.7 3.91 Rue deer 55 0.23 White-tailed deer 165 0.55 American elk 595 0.27 African elephant 8160 0.06 Find a formula that models r as a power function of W, and draw a graph of this function.arrow_forwardMore on the Pacific Sardine This is a continuation of Example 5.1. In this exercise, we explore the Pacific sardine population further, using the model in Example 5.1. a. If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2milliontons? Suggestion: One way to solve this is to make a new logistic formula using K2.4, r0.338, and N(0)0.05. b. The value of r used in Example 5.1 ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year with the carrying capacity still at 2.4milliontons. Answer the question in part a using this lower value of r. Note: The population estimate of 50,000 tons and the adjusted value of r are given in the paper by Murphy see footnote 3 on page 347. Murphy points out that factoring in the growth of the competing anchovy population makes the recovery times even longer, and he adds. "It is disconcerting to realize how slowly the population will recover to its level of maximum productivity ... even if fishing stops." Studies to fit a logistic model to the Pacific sardine population have yielded. N=241+239e0.338t where t is measured in years and N is measured in millions of tons of fish. Part 1 What is r for the Pacific sardine? Part 2 According to the logistic model, in the absence of limiting factors, what would be the annual percentage growth rate for the Pacific sardine? Part 3 What is the environmental carrying capacity K? Part 4 What is the optimum yield level? Part 5 Make a graph of N versus t. Part 6 At what time t should the population he harvested? Part 7 What portion of the graph is concave up? What portion is concave down?arrow_forwardPopulation Growth These exercises use the population growth model. 3. Squirrel Population A grey squirrel population was intro-duced in a certain county or Great Britain 30 years ago. Biologists observe that thepopulation double every 6 years, and now population is 100,000. (a) What was the initial size Of the squirrel population? (b) Estimate the squirrel population 10 years from now. (e) Sketch a graph of the squirrel population.arrow_forward
- Logistic Population growth the table and scatter plot give the population of black flies in a closed laboratory container over an 18 day period. (a) Use the logistic command on your calculator to find a logistic model for these data. (b) Use the model to estimate the time when there were 400 flies in the containerarrow_forwardBird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model n(t)=56000.5+27.5e0.044t where t is measured in years. a Find the initial bird population. b Draw a graph of the function n(t). c What size does the population approach as time goes on?arrow_forwardPopulation Growth These exercises use the population growth model. 4. Bird Population A certain species of bird was introduced in a certain county 25 years ago. Biologists observe that the population doubles every 10 years, and now the population is 13,000. (a) What was initial size Of the bird population? (b) Estimate the bird 5 years from now. (c) Sketch a graph Of the birdarrow_forward
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