Exponential and Logarithmic Functions 1. The exponential function 𝑓(𝑥) = 42.21(1.56)𝑥 models the average amount spent, f(x), in dollars, at a shopping mall after x hours. What is the average amount spent, to the nearest dollar, after three hours at a shopping mall? 2. The exponential function 𝑓(𝑥) = 1066𝑒 0.042𝑥 models the gray wolf population of the Western Great Lakes, f(x), x years after 1978. Project the gray wolf’s population in the recovery area in 2012. 3. After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: For n compounding periods per year: r A P 1 n nt For continuous compounding A Pert a. A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to quarterly compounding. b. A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to continuous compounding. 4. The percentage of adult height attained by a boy who is x years old can be modeled by 𝑓(𝑥) = 29 + 48.8 log(𝑥 + 1) where x represents the boy’s age (from 5 to 15) and f(x) represents the percentage of his adult height. Approximately what percentage of his adult height has a boy attained at age ten? 5. When the outside air temperature is anywhere from 72° to 96° Fahrenheit, the temperature in an enclosed vehicle climbs by 43° in the first hour. The function 𝑓(𝑥) = 13.4 ln 𝑥 − 11.6 models the temperature increase, f(x), in degrees Fahrenheit, after x minutes. Use the function to find the temperature increase, to the nearest degree, after 30 minutes. Exponential and Logarithmic Equations 6. How long, to the nearest tenth of a year, will it take $1000 to grow to $3600 at 8% annual interest compounded quarterly? 7. The percentage of adult height attained by a girl who is x years old can be modeled by 𝑓(𝑥) = 62 + 35 log(𝑥 − 4) where x represents the girl’s age (from 5 to 15) and f(x) represents the percentage of her adult height. At what age has a girl attained 97% of her adult height? 8. In 2000, the population of Africa was 807 million and by 2011 it had grown to 1052 million. Use the exponential growth model A A0e kt , in which t is the number of years after 2000, to find the exponential growth function that models the data. By which year will Africa’s population reach 2000 million, or two billion? 9. The half-life of strontium-90 is 28 years, meaning that after 28 years a given amount of the substance will have decayed to half the original amount. Find the exponential decay model for strontium-90. If there are originally 60 grams, how long will it take for strontium-90 to decay to a level of 10 grams? 0.8 10. In a learning theory project, psychologists discovered that 𝑓(𝑡) = 1+𝑒 −0.2𝑡 is a model for describing the proportion of correct responses, f(t), after t learning trials. Find the proportion of correct responses prior to learning trials taking place.