501 4.6 Applications and Models of Exponential Growth and Decay The constant k determines the type of function. • When k 70, the function describes growth. Examples of exponential growth include compound interest and atmospheric carbon dioxide. • When k 60, the function describes decay. One example of exponential decay is radioactive decay. 4.6 Applications and Models of Exponential Growth and Decay ■ The Exponential Growth or Decay Function ■ Growth Function Models ■ Decay Function Models The Exponential Growth or Decay Function In many situations in ecology, biology, economics, and the social sciences, a quantity changes at a rate proportional to the amount present. The amount present at time t is a special function of t called an exponential growth or decay function. Exponential Growth or Decay Function Let y0 be the amount or number present at time t = 0. Then, under certain conditions, the amount y present at any time t is modeled by y =y0e kt, where k is a constant. LOOKING AHEAD TO CALCULUS The exponential growth and decay function formulas are studied in calculus in conjunction with the topic known as differential equations. EXAMPLE 1 Determining a Function to Model Exponential Growth Earlier in this chapter, we discussed the growth of atmospheric carbon dioxide over time using a function based on the data from the table. Now we determine such a function from the data. (a) Find an exponential function that gives the amount of carbon dioxide y in year t. (b) Estimate the year when future levels of carbon dioxide will be double the preindustrial level of 280 ppm. SOLUTION (a) The data points exhibit exponential growth, so the equation will take the form y = y0e kt. We must find the values of y0 and k. The data begin with the year 1990, so to simplify our work we let 1990 correspond to t = 0, 1991 correspond to t = 1, and so on. Here y0 is the initial amount and y0 = 353 in 1990 when t = 0. Thus the equation is y = 353ekt. Let y 0 = 353. From the last pair of values in the table, we know that in 2275 the carbon dioxide level is expected to be 2000 ppm. The year 2275 corresponds to 2275 - 1990 = 285. Substitute 2000 for y and 285 for t, and solve for k. Year Carbon Dioxide (ppm) 1990 353 2000 375 2075 590 2175 1090 2275 2000 Data from International Panel on Climate Change (IPCC). Growth Function Models The amount of time it takes for a quantity that grows exponentially to become twice its initial amount is its doubling time.
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