Use Exponential Models in Applications In the following exercises, solve. In the last five years the population of the United States has grown at a rate of 0.7% per year to about 318,900,000. If this rate continues, what will be the population in 5 more years?
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Use Exponential Models in Applications
In the following exercises, solve.
In the last five years the population of the United
States has grown at a rate of 0.7% per year to about
318,900,000. If this rate continues, what will be the
population in 5 more years?
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- The rate of U.S. per capita sales of bottled water for the period 2000–2010 can be approximated by s(t) = −0.18t2 + 3t + 15 gallons per year (0 ≤ t ≤ 10), where t is the time in years since the start of 2000.† After conducting a survey of sales in your state, you estimate that consumption in gyms accounts for a fraction f(t) = 0.2 + 0.04t of all bottled water consumed in year t. Assuming that your model is correct, estimate, to the nearest gallon, the total amount of bottled water consumed per capita in gyms from the start of 2002 to the start of 2010. HINT [Rate of consumption = s(t)f(t).] _____galSediment buildup behind a dam can adversely affect a reservoir's storage capacity. Measurements of a reservoir in 1988 showed it had a capacity of 500,000 acre-feet of water. Measurements in 2018 showed that the reservoir's capacity has been decreasing at a rate of 2,000 acre-feet per year. Find a linear model of the number of acre-feet A of water in the reservoir n years after 1988.The circulation C of a certain magazine as a function of time t is given by the formula c=5.2/0.1+0.3t Here C is measured in thousands and t is measured in years since the beginning of 2006 when the magazine was started. Find when the circulation of the magazines will reach 34.83 thousand. Write the equation you are solving.
- In the December, 1969, American Economic Review (pp. 886-896), Nathanial Leff reports thefollowing least squares regression results for a cross section study of the effect of age composition onsavings in 74 countries in 1964:log S/Y = 7.3439 + 0.1596 log Y/N + 0.0254 log G - 1.3520 log D1 - 0.3990 log D2 (R2= 0.57)log S/N = 8.7851 + 1.1486 log Y/N + 0.0265 log G - 1.3438 log D1 - 0.3966 log D2 (R2= 0.96)where S/Y = domestic savings ratio, S/N = per capita savings, Y/N = per capita income, D1 = percentage ofthe population under 15, D2 = percentage of the population over 64, and G = growth rate of per capitaincome. Are these results correct? Explain..The population of a town has been growing, following the equation P=800t+5500P=800t+5500, where t is years after 2010. The number of restaurants in the town has been growing according to the equation R=2t+40R=2t+40.Complete an equation for the number of restaurants per capita (per person)Restaurants per capita: How many restaurants per capita does this model predict for the year 2017?U.S. Population The projected population of theUnited States for selected years from 2000 to 2060 isshown in the table below, with the population givenin millions.a. According to this table, what should the U.S. population have been in 2010?b. Create a new table with x representing the numberof years after 2000 and y representing the numberof millions.c. Use a graphing utility to graph the data from thenew table as a scatter plot.
- A researcher has a sample of 6 annual observations {94, 104, 102, 99, 111 and 107} for the CPI in country Z for the period 2015 to 2020, and wants to forecast CPI for the years 2021, 2022 and 2023. The researcher uses 3 different forecasting models: A, B and C. Model A is an AR(1) model with no drift and with an estimated autoregressive coefficient = 0.7. Model B is a MA(1) model with no constant and with an estimated MA coefficient = -0.4 (note the minus !). Model C is a random walk model with no drift. The error terms over the 2015-2020 period were estimated to have the values: {3, -1, 2, 4, -3, 1}. a. Compute the 2021, 2022 and 2023 forecasted values for the consumer price index based on the three models. Show the formulas and the details of your calculations, and explain all the related symbols. b. Suppose that the actual values of the CPI over the 2021, 2022 and 2023 were {108, 114, 105}. Calculate the Root mean square error of the three model forecasts over the 2021-2023…A town has a population of 5000 persons, but is expected to grow by 2% every year. (i) What wound be the population size in 7 years? (ii) Find the sum of the first eight terms of the sequence 1/8,−1/4 ,1/2,.... Please show working.The forecasting staff for the Prizer Corporation has developed a model to predict sales of its air-cushioned-ride snowmobiles. The model specifies that sales S vary jointly with disposable personal income Y and the population between ages 15 and 40, Z, and inversely with the price of the snowmobiles P. Based on past data, the best estimate of this relationship is S=kYZP where k has been estimated (with past data) to equal 100. If Y=11,000,Z=1,200, and P=20,000, what value would you predict for S? What happens if P is reduced to $17,500? How would you go about developing a value for k? What are the potential weaknesses of this model?
- Logarithms are especially useful for comparing series with two divergent scales since 10 percent growth always looks the same, regardless of the starting level. When absolute levels matter, the raw data are more appropriate, but when growth rates are whats important, log scales are better.Given the regression equationY = -50 + 12Xa. What is the change in Y when X changes by +3?b. What is the change in Y when X changes by -4?c. What is the predicted value of Y when X = 12?d. What is the predicted value of Y when X = 23?e. Does this equation prove that a change in X causes a change in Y?(Ch7) True or False? We should use a normal distribution to model the number of years until the next Florida hurricane strike.