503 4.6 Applications and Models of Exponential Growth and Decay Decay Function Models Half-life is the amount of time it takes for a quantity that decays exponentially to become half its initial amount. EXAMPLE 3 Using an Exponential Function to Model Population Growth According to the U.S. Census Bureau, the world population reached 6 billion people during 1999 and was growing exponentially. By the middle of 2018, the population had grown to 7.504 billion. The projected world population (in billions of people) t years after 2018 is given by the function ƒ1t2 = 7.504e0.00936t. (a) Based on this model, what will the world population be in 2025? (b) If this trend continues, approximately when will the world population reach 9 billion? SOLUTION (a) Since t = 0 represents the year 2018, in 2025, t would be 2025 - 2018 = 7 yr. We must find ƒ1t2 when t is 7. ƒ1t2 = 7.504e0.00936t Given function ƒ172 = 7.504e0.00936172 Let t = 7. ƒ172 ≈8.012 Use a calculator. Based on this model, the population will be 8.012 billion in 2025. (b) ƒ1t2 = 7.504e0.00936t Given function 9 = 7.504e0.00936t Let ƒ1t2 = 9. 9 7.504 = e0.00936t Divide by 7.504. ln 9 7.504 = ln e0.00936t Take the natural logarithm on each side. ln 9 7.504 = 0.00936t ln ex = x, for all x. t = ln 9 7.504 0.00936 Divide by 0.00936 and rewrite. t ≈19.4 Use a calculator. According to this model, 19.4 yr after 2018, which would be during the year 2037, world population will reach 9 billion. S Now Try Exercise 39. NOTE In Example 4 on the next page, the initial amount of substance is given as 600 g. Because half-life is constant over the lifetime of a decaying quantity, starting with any initial amount, y0 , and substituting 1 2 y0 for y in y = y0 ek t would allow the common factor y 0 to be divided out. The rest of the work would be the same.
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