In 2930, a quantity P is an exponential function of time t. Use the given information

For Problems 116, solve for t using natural logarithms. 10 = 2t

For Problems 116, solve for t using natural logarithms. 5t = 7

For Problems 116, solve for t using natural logarithms. 2 = (1.02)t

For Problems 116, solve for t using natural logarithms. 130 = 10t

For Problems 116, solve for t using natural logarithms. 10 = et

For Problems 116, solve for t using natural logarithms. 100 = 25(1.5)t

For Problems 116, solve for t using natural logarithms. 50 = 10 3t

For Problems 116, solve for t using natural logarithms. 5 = 2et

For Problems 116, solve for t using natural logarithms. e3t = 100 1

For Problems 116, solve for t using natural logarithms. 10 = 6e0.5t 1

For Problems 116, solve for t using natural logarithms. 40 = 100e0.03t 1

For Problems 116, solve for t using natural logarithms. a = bt 1

For Problems 116, solve for t using natural logarithms. B = Pert 1

For Problems 116, solve for t using natural logarithms. 2P = Pe0.3t 1

For Problems 116, solve for t using natural logarithms. 7 3t = 5 2t 1

For Problems 116, solve for t using natural logarithms. 5e3t = 8e2t 1

The functions in Problems 1720 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. P = 5(1.07)t 1

The functions in Problems 1720 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. P = 7.7(0.92)t 1

The functions in Problems 1720 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. P = 15e0.06t 2

The functions in Problems 1720 represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. P = 3.2e0.03t 2

Write the functions in Problems 2124 in the form P = P0at. Which represent exponential growth and which represent exponential decay? P = 15e0.25t 2

Write the functions in Problems 2124 in the form P = P0at. Which represent exponential growth and which represent exponential decay? P = 2e0.5t 2

Write the functions in Problems 2124 in the form P = P0at. Which represent exponential growth and which represent exponential decay? P = P0e0.2t 2

Write the functions in Problems 2124 in the form P = P0at. Which represent exponential growth and which represent exponential decay? P = 7et 2

In Problems 2528, put the functions in the form P = P0ekt. P = 15(1.5)t 2

In Problems 2528, put the functions in the form P = P0ekt. P = 10(1.7)t 2

In Problems 2528, put the functions in the form P = P0ekt. P = 174(0.9)t 2

In Problems 2528, put the functions in the form P = P0ekt. P = 4(0.55)t 2

In Problems 2930, a quantity P is an exponential function of time t. Use the given information about the function P = P0ekt to: (a) Find values for the parameters k and P0. (b) State the initial quantity and the continuous percent rate of growth or decay. P = 140 when t = 3 and P = 100 when t = 1 3

In Problems 2930, a quantity P is an exponential function of time t. Use the given information about the function P = P0ekt to: (a) Find values for the parameters k and P0. (b) State the initial quantity and the continuous percent rate of growth or decay. P = 40 when t = 4 and P = 50 when t = 3 3

(a) What is the continuous percent growth rate for P = 100e0.06t, with time, t, in years? (b) Write this function in the form P = P0at. What is the annual percent growth rate? 3

(a) What is the annual percent decay rate for P = 25(0.88)t, with time, t, in years? (b) Write this function in the form P = P0ekt. What is the continuous percent decay rate? 3

What annual percent growth rate is equivalent to a continuous percent growth rate of 8%? 3

What continuous percent growth rate is equivalent to an annual percent growth rate of 10%? 3

The following formulas give the populations of four different towns, A, B, C, and D, with t in years from now. PA = 600e0.08t PB = 1000e0.02t PC = 1200e0.03t PD = 900e0.12t (a) Which town is growing fastest (that is, has the largest percentage growth rate)? (b) Which town is the largest now? (c) Are any of the towns decreasing in size? If so, which one(s)? 3

A citys population is 1000 and growing at 5% a year. (a) Find a formula for the population at time t years from now assuming that the 5% per year is an: (i) Annual rate (ii) Continuous annual rate (b) In each case in part (a), estimate the population of the city in 10 years. 3

The population, P, inmillions, ofNicaraguawas 5.4 million in 2004 and growing at an annual rate of 3.4%. Let t be time in years since 2004. (a) Express P as a function in the form P = P0at. (b) Express P as an exponential function using base e. (c) Compare the annual and continuous growth rates. 3

The gross world product is W = 32.4(1.036)t, where W is in trillions of dollars and t is years since 2001. Find a formula for gross world product using a continuous growth rate. 3

The population of the world can be represented by P = 7(1.0115)t, where P is in billions of people and t is years since 2012. Find a formula for the population of the world using a continuous growth rate. 4

A fishery stocks a pondwith 1000 young trout. The number of trout t years later is given by P(t) = 1000e0.5t . (a) How many trout are left after six months? After 1 year? (b) Find P(3) and interpret it in terms of trout. (c) At what time are there 100 trout left? (d) Graph the number of trout against time, and describe how the population is changing. What might be causing this? 4

The Hershey Company is the largest US producer of chocolate. In 2011, annual net sales were 6.1 billion dollars and were increasing at a continuous rate of 4.2% per year.65 (a) Write a formula for annual net sales, S, as a function of time, t, in years since 2011. (b) Estimate annual net sales in 2015. (c) Use a graph to estimate the year in which annual net sales are expected to pass 8 billion dollars and check your estimate using logarithms. 4

During a recession a firms revenue declines continuously so that the revenue, R (measured in millions of dollars), in t years time is given by R = 5e0.15t. (a) Calculate the current revenue and the revenue in two years time. (b) After how many years will the revenue decline to $2.7 million? 4

The population of a city is 50,000 in 2008 and is growing at a continuous yearly rate of 4.5%. (a) Give the population of the city as a function of the number of years since 2008. Sketch a graph of the population against time. (b) What will be the citys population in the year 2018? (c) Calculate the time for the population of the city to reach 100,000. This is called the doubling time of the population. 4

For children and adults with diseases such as asthma, the number of respiratory deaths per year increases by 0.33% when pollution particles increase by a microgram per cubic meter of air.66 (a) Write a formula for the number of respiratory deaths per year as a function of quantity of pollution in the air. (Let Q0 be the number of deaths per year with no pollution.) (b) What quantity of air pollution results in twice as many respiratory deaths per year as there would be without pollution? 4

The concentration of the car exhaust fume nitrous oxide, NO2, in the air near a busy road is a function of distance from the road. The concentration decays exponentially at a continuous rate of 2.54% per meter.67 At what distance from the road is the concentration of NO2 half what it is on the road? 4

With time, t, in years since the start of 1980, textbook prices have increased at 6.7% per year while inflation has been 3.3% per year.68 Assume both rates are continuous growth rates. (a) Find a formula for B(t), the price of a textbook in year t if it cost $B0 in 1980. (b) Find a formula for P(t), the price of an item in year t if it cost $P0 in 1980 and its price rose according to inflation. (c) A textbook cost $50 in 1980. When is its price predicted to be double the price that would have resulted from inflation alone? 4

In 2011, the populations of China and India were approximately 1.34 and 1.19 billion people69, respectively. However, due to central control the annual population growth rate of China was 0.4% while the population of India was growing by 1.37% each year. If these growth rates remain constant, when will the population of India exceed that of China? 4

In 2010, there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US.70 The number of vehicles grew 15.5% over the previous decade, while the population has been growing at 9.7% per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?