Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Male and Female High School Graduates The table below shows the percentage of male and female high school graduates who enrolled in college within 12 months of graduation.
Years | 1960 | 1965 | 1970 | 1975 |
Males | 54% | 57.3% | 55.2% | 52.6% |
Females | 37.9% | 45.3% | 48.5% | 49% |
a. Find the equation of the regression line for percentage of male high school graduates entering college as a function of time.
b. Find the equation of the regression line for percentage of female high school graduates entering college as a function of time.
c. Assume that the regression lines you found in part a and part b represent trends in the data. If the trends persisted, when would you expect first to have seen the same percentage of female and male graduates entering college? (You may be interested to know that this actually occurred for the first time in 1980. The percentages fluctuated but remained very close during the 1981s and 1990s. In the 2000s, more female graduates entered college than did males. In 2008, for example, the rate for males was 66% compared with 72% for females.)
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. High School GraduatesThe following table shows the number, in millions, graduating from high school in the United States in the given year. Year Number 1975 3.13 1980 3.04 1985 2.68 1990 2.57 1994 2.46 1999 2.76 2004 3.05 2009 3.32 a.Make a plot of the data and explain why a linear model is not appropriate. b.Use regression to find a linear model for the years 1975 through 1990. In this part and the next. round regression line parameters to three decimal places. c.Use regression to find a linear model for the years 1994 through 2009. d.Write a formula for a model of the number, in millions, graduating as a piecewise-defined function using the linear models from part b and part c. e.Make a graph of the formula you found in part d. f.The number graduating in 1995 was 2.52million. On the basis of your graph in part e, determine how this compares with what would be expected from your formula.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. An Inappropriate Linear Model for Radioactive Decay This is a continuation of Exercise 14. Physicists have established that radioactive substances display constant percentage decay, and thus radioactive decay is appropriately modeled exponentially. This exercise is designed to show how using data without an understanding of the phenomenon that generated them can lead to inaccurate conclusions. a. Plot the data points from Exercise 14. Do they appear almost to fall on a straight line? b. Find the equation of the regression line and add its graph to the one you made in part a. c. If you used the regression line as a model for decay of 239U, how long would it take for the Initial 1 gram to decay to half that amount? Compare this with your answer to part d of Exercise 14. d. The linear model represented by the regression line makes an absurd prediction concerning the amount of uranium Uranium-239 remaining after 1 hour. What is this prediction? 14. The half life of 239U Uranium-239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of 239U was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below. Time, in minutes Grams remaining 0 1 1 0.971 2 0.943 3 0.916 4 0.889 5 0.863 a. Show that these are exponential data and find an exponential model For this problem, round all your answers to three decimal places. b. What is the percentage decay rate each minute? What does this number mean in practical terms? c. Use functional notation to express the amount remaining after 10 minutes and then calculate the value. d. What is the half life of 239U?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Telecommunications The following table shows the annual expenditures, in dollars, per customer unit for residential landline phone services and cellular phone services in the United States in the given year. Year Landline Cell 2004 592 378 2006 542 524 2008 467 643 2010 401 760 Calculate the regression line for each type of service, and determine expenditure level at which the two line cross. Round your answer for the expenditure level to one decimal place.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Quarterly Pine Pulpwood PricesIn southwest Georgia, the average pine pulpwood prices vary predictably over the course of the year, primarily because of weather. Prices in 2009 followed this pattern. At the beginning of the first quarter, the average price P was 9 per ton. During the first quarter, prices declined steadily to 8 per ton, then remained steady at 8 per ton through the end of the third quarter. During the fourth quarter, prices increased steadily from 8 to 10 per ton. a.Sketch a graph of pulpwood prices as a function of the quarter in the year. b.What formula for price P as a function of t, the quarter, describes the price from the beginning of the year through the first quarter? c.What formula for price P as a function of t, the quarter, describes the price from the first to the third quarter? d.What formula for price P as a function of t, the quarter, describes the price from the third to the fourth quarter? e.Write a formula for price P throughout the year as a piecewise-defined function of t, the quarter.arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Running Ants A scientist collected the following data on the speed, in centimeters per second, at which ants ran at the given ambient temperature, in degrees Celsius. Temperature Speed 25.6 2.62 27.5 3.03 30.3 3.57 30.4 3.56 32.2 4.03 33.0 4.17 33.8 4.32 a.Find the equation of the regression line, giving the speed as a function of the temperature. b.Explain in practical terms the meaning of the slope of the regression line. c.Express, using functional notation, the speed at which the ants run when the ambient temperature is 29 degrees Celsius, and then estimate that value. d.The scientist observes the ants running at a speed of 2.5 centimeters per second. What is the ambient temperature?arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. Is a Linear Model Appropriate? The number, in thousands, of bacteria in a petri dish is given by the following table. Time is measured in hours. Time in hours since experiment began Number of bacteria in thousands 0 1.2 1 2.4 2 4.8 3 9.6 4 19.2 5 38.4 6 76.8 The table below shows enrollment, in millions of people, in private colleges in the United States during the years from 2004 through 2008. Date Enrollment in millions 2004 4.29 2005 4.47 2006 4.58 2007 4.76 2008 5.13 a.Plot the data points for number of bacteria. Does it look reasonable to approximate these data with a straight line? b.Plot the data points for college enrollment. Does it look reasonable to approximate these data with a straight line?arrow_forward
- ReminderRound all answers to two decimal places unless otherwise indicated. TargetData from Targets 2014 annual report indicate that the equation of change for the revenue R, in millions of dollars, from 2010 through 2014 is dRdt=1647.7 where t is the time, in years, since 2010. If the initial revenue is 66,726.4 million dollars, find an equation that gives R as a linear function of t.arrow_forwardRemainder Round all answers to two decimal places unless otherwise indicated. 2. Federal Methamphetamine Arrests The table below shows the number A, in thousands, of federal arrests for methamphetamine t years after 2006. t = years since 2006 A= thousands of arrests 0 5.85 1 5.54 2 4.72 3 4.70 Find the equation of the regression line for A as a function of t.arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Gasoline Prices In 1960, the average price per gallon of gasoline was 31 cents per gallon. Form 1960 to 2000, prices increased, on average, by 2.5 cents per gallon per year. 4 a. Using G for the price, in cents per gallon, and t for the time, in years, since 1960, use a formula to express G as linear function of t. b. What price per gallon does the model yield for 1990? Note: The actual price was 1.00 per gallon. c. Use the Internet to find the average price of gasoline for the current year. Does the model from part a give a price near the current price?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Chemical Reaction The following table shows for a certain chemical reaction, the rate of reaction R, in moles per second, as a function of the concentration x, in moles per cubic meter, of the product. Concentration x 10 20 30 40 50 Reaction rate R 18 12 7 3 0 a. Use quadratic regression to find a model for the data. Round regression parameters to three decimal places. b. Use your model to estimate R(24), and explain what your answer means. c. Estimate the concentration at which the reaction rate is 6 moles per cubic meter per second. Consider concentrations only up to a level of 50moles per cubic meter.arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Budget Constraints Your family likes to eat fruit, but because of budget constraints, you spend only 5 each week on fruit. Your two choices are apples and grapes. Apples cost 1 per pound, and grapes cost 2 per pound. Let a denote the number f pounds of apples you buy and g the number of pounds of grapes. Because of your budget, it is possible to express g as a linear function of the variable a. To find the linear formula, we need to find its slope and initial value. a. If you buy one more pound of apples, how much less money do you have available to spend on grapes? Then how many fewer pounds of grapes can you buy? b. Use your answer to part a to find the slope of g as a linear function of a. Hint: Remember that the slope is the change in the function that results from increasing the variable by 1. Should the slope of g be positive or negative? c. To find the initial value of g, determine how many pounds of grapes you can buy if you buy no apples. d. Use your answer to parts b and c to find a formula for g as a linear function of a.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Magazine SalesThe following table shows the income from sales of a certain magazine, measured in thousands of dollars, at the start of the given year. Year Income 2009 7.76 2010 8.82 2011 9.88 2012 10.94 2013 12.00 2014 13.08 2015 14.26 2016 15.54 Over an initial period the sales grew at a constant rate, and over the rest of the time the sales grew at a constant percentage rate. Calculate differences and ratios to determine what these time periods are, and find the growth rate or percentage growth rate, as appropriate.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning