In 2002, a company had 80 retail stores. By 2008, the company had grown to 180 retail stores. If the company is growing exponentially, find a formula for the function. By defining our input variable to be t, years after 2002, the information listed can be written as two input-output pairs: (0,80) and (6,180). Notice that by choosing our input variable to be measured as years after the first year value provided, we have effectively “given” ourselves the initial value for the function: a = 80. This gives us an equation of the form t f (t) 80b . Substituting in our second input-output pair allows us to solve for b: 6 180 80 b Divide by 80 6 180 9 80 4 b Take the 6th root of both sides. 6 9 1.1447 4 b This gives us our equation for the population: t f (t) 80(1.1447) Recall that since b = 1+r, we can interpret this to mean that the population growth rate is r = 0.1447, and so the population is growing by about 14.47% each year. 190 Chapter 5 In this example, you could also have used (9/4)^(1/6) to evaluate the 6th root if your calculator doesn’t have an nth root button. We chose to use the x f ( form of the x) ab exponential function rather than the x f (x) a(1 r) form, but this choice was entirely arbitrary – either form would be fine to use. When finding equations, the value for b or r will usually have to be rounded to be written easily. To preserve accuracy, it is important to not over-round these values. Typically, you want to be sure to preserve at least 3 significant digits in the growth rate. For example, if your value for b was 1.00317643, you would want to round this no further than to 1.00318.
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