In 2023, a quantity P is an exponential function of time t. Use the given information

The following functions give the populations of four towns with time t in years. (i) P = 600(1.12)t (ii) P = 1,000(1.03)t (iii) P = 200(1.08)t (iv) P = 900(0.90)t (a) Which town has the largest percent growth rate? What is the percent growth rate? (b) Which town has the largest initial population? What is that initial population? (c) Are any of the towns decreasing in size? If so, which one(s)?

Each of the following functions gives the amount of a substance present at time t. In each case, give the amount present initially (at t = 0), state whether the function represents exponential growth or decay, and give the percent growth or decay rate. (a) A = 100(1.07)t (b) A = 5.3(1.054)t (c) A = 3500(0.93)t (d) A = 12(0.88)t

Figure 1.64 shows graphs of several cities populations against time. Match each of the following descriptions to a graph and write a description to match each of the remaining graphs. (a) The population increased at 5% per year. (b) The population increased at 8% per year. (c) The population increased by 5000 people per year. (d) The population was stable. I II III IV V VI time (years) population Figure 1.64

Give a possible formula for the functions in Problems 45.

Give a possible formula for the functions in Problems 45.

The gross domestic product, G, of Switzerland was 310 billion dollars in 2007. Give a formula for G (in billions of dollars) t years after 2007 if G increases by (a) 3%peryear (b) 8 billion dollars per year

A town has a population of 1000 people at time t = 0. In each of the following cases, write a formula for the population, P, of the town as a function of year t. (a) The population increases by 50 people a year. (b) The population increases by 5% a year.

A product costs $80 today. How much will the product cost in t days if the price is reduced by (a) $4aday (b) 5%aday

An air-freshener starts with 30 grams and evaporates. In each of the following cases, write a formula for the quantity, Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. The decrease is: (a) 2gramsaday (b) 12% a day 1

World population is approximately P = 6.4(1.0126)t , with P in billions and t in years since 2004. (a) What is the yearly percent rate of growth of the world population? (b) What was the world population in 2004? What does this model predict for the world population in 2010? (c) Use part (b) to find the average rate of change of the world population between 2004 and 2010. 1

A 50 mg dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of 6% per hour. (a) Find a formula for the amount, A (in mg), of quinine in the body t hours after the dose is given. (b) How much quinine is in the body after 24 hours? (c) Graph A as a function of t. (d) Use the graph to estimate when 5 mg of quinine remains. 1

The consumer price index (CPI) for a given year is the amount of money in that year that has the same purchasing power as $100 in 1983. At the start of 2009, the CPI was 211. Write a formula for the CPI as a function of t years after 2009, assuming that the CPI increases by 2.8% every year. 1

During the 1980s, Costa Rica had the highest deforestation rate in the world, at 2.9% per year. (This is the rate at which land covered by forests is shrinking.) Assuming the rate continues, what percent of the land in Costa Rica covered by forests in 1980 will be forested in 2015? 1

Graph y = 100e0.4x. Describe what you see. 1

(a) Make a table of values for y = ex using x = 0, 1, 2, 3. (b) Plot the points found in part (a). Does the graph look like an exponential growth or decay function? (c) Make a table of values for y = ex using x = 0, 1, 2, 3. (d) Plot the points found in part (c). Does the graph look like an exponential growth or decay function? 1

For Problems 1617, find a possible formula for the function represented by the data. x 0 1 2 3 f(x) 4.30 6.02 8.43 11.80 1

For Problems 1617, find a possible formula for the function represented by the data. t 0 1 2 3 g(t) 5.50 4.40 3.52 2.82 1

In Problems 1819, find all the tables that have the given characteristic. y could be a linear function of x. 1

In Problems 1819, find all the tables that have the given characteristic. y could be an exponential function of x. 2

In Problems 2023, a quantity P is an exponential function of time t. Use the given information about the function P = P0at to: (a) Find values for the parameters a and P0. (b) State the initial quantity and the percent rate of growth or decay. P0a3 = 75 and P0a2 = 50 2

In Problems 2023, a quantity P is an exponential function of time t. Use the given information about the function P = P0at to: (a) Find values for the parameters a and P0. (b) State the initial quantity and the percent rate of growth or decay. P0a4 = 18 and P0a3 = 20 2

In Problems 2023, a quantity P is an exponential function of time t. Use the given information about the function P = P0at to: (a) Find values for the parameters a and P0. (b) State the initial quantity and the percent rate of growth or decay. P = 320 when t = 5 and P = 500 when t = 3 2

In Problems 2023, a quantity P is an exponential function of time t. Use the given information about the function P = P0at to: (a) Find values for the parameters a and P0. (b) State the initial quantity and the percent rate of growth or decay. P = 1600 when t = 3 and P = 1000 when t = 1 2

If the worlds population increased exponentially from 5.937 billion in 1998 to 6.771 billion in 2008 and continued to increase at the same percentage rate between 2008 and 2012, calculate what the worlds population would have been in 2012. How does this compare to the Population Reference Bureau estimate of 7.07 billion in July 2012? 58 2

The number of passengers using a railway fell from 190,205 to 174,989 during a 5-year period. Find the annual percentage decrease over this period. 2

The company that produces Cliffs Notes (abridged versions of classic literature) was started in 1958 with $4000 and sold in 1998 for $14,000,000. Find the annual percent increase in the value of this company over the 40 years. 2

Find a formula for the number of zebra mussels in a bay as a function of the number of years since 2010, given that there were 2700 at the start of 2010 and 3186 at the start of 2011. (a) Assume that the number of zebra mussels is growing linearly. Give units for the slope of the line and interpret it in terms of zebra mussels. (b) Assume that the number of zebra mussels is growing exponentially.What is the annual percent growth rate of the zebra mussel population? 2

Worldwide, wind energy59 generating capacity, W, was 39,295 megawatts in 2003 and 120,903 megawatts in 2008. (a) Use the values given to write W, in megawatts, as a linear function of t, the number of years since 2003. (b) Use the values given to write W as an exponential function of t. (c) Graph the functions found in parts (a) and (b) on the same axes. Label the given values. (d) Use the functions found in parts (a) and (b) to predict the wind energy generated in 2010. The actual wind energy generated in 2010 was 196,653 megawatts. Comment on the results: Which estimate is closer to the actual value? 2

(a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions. 3

Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an exponential function, find a formula for the function. (a) x f(x) 0 10.5 1 12.7 2 18.9 3 36.7 (b) t s(t) 1 50.2 0 30.12 1 18.072 2 10.8432 (c) u g(u) 0 27 2 24 4 21 6 18 3

(a) Could the data on annual world soybean production 60 in Table 1.34 correspond to a linear function or an exponential function? If so, which? (b) Find a formula for P, world soybean production in millions of tons, as a function of time, t, in years since 2000. (c) What is the annual percent increase in soybean production? Table 1.34 Soybean production, in millions of tons Year 2000 2001 2002 2003 2004 2005 Production 161.0 170.3 180.2 190.7 201.8 213.5 3

The 2004 US presidential debates questioned whether the minimum wage has kept pace with inflation. Decide the question using the following information:61 In 1938, the minimum wage was 25c/; in 2004, it was $5.15. During the same period, inflation averaged 4.3%. 3

(a) Niki invested $10,000 in the stock market. The investment was a loser, declining in value 10% per year each year for 10 years. How much was the investment worth after 10 years? (b) After 10 years, the stock began to gain value at 10% per year. After how long will the investment regain its initial value ($10,000)? 3

A photocopy machine can reduce copies to 80% of their original size. By copying an already reduced copy, further reductions can be made. (a) If a page is reduced to 80%, what percent enlargement is needed to return it to its original size? (b) Estimate the number of times in succession that a page must be copied to make the final copy less than 15% of the size of the original. 3

Whooping cough was thought to have been almost wiped out by vaccinations. It is now known that the vaccination wears off, leading to an increase in the number of cases, w, from 1248 in 1981 to 18,957 in 2004. (a) With t in years since 1980, find an exponential function that fits this data. (b) What does your answer to part (a) give as the average annual percent growth rate of the number of cases? (c) On May 4, 2005, the Arizona Daily Star reported (correctly) that the number of cases had more than doubled between 2000 and 2004. Does your model confirm this report? Explain. 3

Aircraft require longer takeoff distances, called takeoff rolls, at high altitude airports because of diminished air density. The table shows how the takeoff roll for a certain light airplane depends on the airport elevation. (Takeoff rolls are also strongly influenced by air temperature; the data shown assume a temperature of 0 C.) Determine a formula for this particular aircraft that gives the takeoff roll as an exponential function of airport elevation. Elevation (ft) Sea level 1000 2000 3000 4000 Takeoff roll (ft) 670 734 805 882 967 3

(a) According to the US Department of Energy, the US consumed 91 million gallons of biodiesel in 2005. Approximately how much biodiesel (in millions of gallons) did the US consume in 2006? In 2007? (b) Graph the points showing the annual US consumption of biodiesel, in millions of gallons of biodiesel, for the years 2005 to 2009. Label the scales on the horizontal and vertical axes. 3

(a) True or false: The annual US consumption of biodiesel grew exponentially from 2003 to 2005. Justify your answer without doing any calculations. (b) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least double? (c) According to this data, during what single year(s), if any, did the US consumption of biodiesel at least triple? 3

Hydroelectric power is electric power generated by the force of moving water. The table shows the annual percent change in hydroelectric power consumption by the US industrial sector.63 Year 2005 2006 2007 2008 2009 % growth over previous yr 1.9 10 45.4 5.1 11 (a) According to the US Department of Energy, the US industrial sector consumed about 29 trillion BTUs of hydroelectric power in 2006. Approximately how much hydroelectric power (in trillion BTUs) did the US consume in 2007? In 2005? (b) Graph the points showing the annual US consumption of hydroelectric power, in trillion BTUs, for the years 2004 to 2009. Label the scales on the horizontal and vertical axes. (c) According to this data, when did the largest yearly decrease, in trillion BTUs, in the US consumption of hydroelectric power occur? What was this decrease? 4

(a) According to the US Department of Energy, the US consumption of wind power was 341 trillion BTUs in 2007. Howmuchwind power did theUSconsume in 2006? In 2008? (b) Graph the points showing the annual US consumption of wind power, in trillion BTUs, for the years 2005 to 2009. Label the scales on the horizontal and vertical axes. (c) Based on this data, in what year did the largest yearly increase, in trillion BTUs, in the US consumption of wind power occur? What was this increase? 4

(a) According to Figure 1.65, during what single year(s), if any, did the US consumption of wind power energy increase by at least 40%? Decrease by at least 40%? (b) Did the US consumption of wind power energy double from 2006 to 2008?