5.6 Applications and Models: Growth and Decay; and Compound Interest Solve applied problems involving exponential growth and decay. Solve applied problems involving compound interest.

Population Growth The function P(t) = P0 ekt, k > 0 is a model of many kinds of population growth. In this function: P0 = population at time 0, P(t) = population after time t, t = amount of time, k = exponential growth rate. The growth rate unit must be the same as the time unit. 2

Example In 2009, the population of Mexico was about 111.2 million, and the exponential growth rate was 1.13% per year. a) Find the exponential growth function. b) Estimate the population in 2014. 3

Example (continued) After how long will the population be double what it was in 2009? 4

Interest Compound Continuously The function P(t) = P0ekt can be used to calculate interest that is compounded continuously. In this function: P0 = amount of money invested, P(t) = balance of the account after t years, t = years, k = interest rate compounded continuously. 5

Example a. What is the interest rate? Suppose that $2000 is invested at interest rate k, compounded continuously, and grows to $2504.65 after 5 years. a. What is the interest rate? 6

Example (continued) b. Find the exponential growth function. c. What will the balance be after 10 years? 7

d. After how long will the $2000 have doubled? 8

Growth Rate and Doubling Time The growth rate k and doubling time T are related by kT = ln 2 or or The population of Kenya is now doubling every 25.8 years. What is the exponential growth rate? 9

Exponential Decay Decay, or decline, of a population is represented by the function P(t) = P0ekt, k > 0. In this function: P0 = initial amount of the substance (at time t = 0), P(t) = amount of the substance left after time t t = time, k = decay rate. The half-life is the amount of time it takes for a substance to decay to half of the original amount. 10

Graphs 11

Decay Rate and Half-Life The decay rate k and the half-life T are related by kT = ln 2 or or Note that the relationship between decay rate and half-life is the same as that between growth rate and doubling time. 12

Example Carbon Dating. The radioactive element carbon-14 has a half-life of 5750 years. The percentage of carbon-14 present in the remains of organic matter can be used to determine the age of that organic matter. Archaeologists discovered that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of its carbon-14 at the time it was found. How old was the linen wrapping? 13

Example (continued) Solution: First find k when the half-life T is 5750 yr: 14

Example (continued) If the linen wrapping lost 22.3% of its carbon-14 from the initial amount P0, then 77.7% is the amount present. To find the age t of the wrapping, solve for t: 15