At the beginning of 1990, 21.7 million people lived in the metropolitan area of a particular city, and the population was growing exponentially. The 1996 population was 25 million. If this trend continues, how large will the population be in the year 2010
Accepted Solution
A:
Answer:approximately 27.5 millionStep-by-step explanation:If 1990 is the initial year, we will rename it as 0. This is the x coordinate in a pair we will need to write the equation that models this particular situation. The y coordinate that goes along with it is 21.7 (x is time in years, y is number of people). The next coordinate pair we have is (6, 25). If 1990 is year 0, 1996 is year 6. The standard form for an exponential equation is[tex]y=a(b)^x[/tex]where y is the number of people, x is the number of years gone by, a is the initial value, and b is the growth rate. We fill in equation 1 with the x and y coordinates from coordinate pair (0, 21.7):[tex]21.7=a(b)^0[/tex]andything rised to the power of 0 = 1, so b raised to 0 = 1:21.7 = a(1) soa = 21.7Now we use coordinate pair (6, 25) in equation 2, subbing in our value for a also:[tex]25=21.7(b)^6[/tex]Divide both sides by 21.7 to get[tex]1.152073733=b^6[/tex]We "undo" that power of 6 by taking the 6th root of both sides:[tex](1.152073733)^{\frac{1}{6}} =(b^6)^{\frac{1}{6}}[/tex]That gives you thatb = 1.0238 (rounded).Now that we have a and b, we can write the model for this situation:[tex]y=21.7(1.0238)^x[/tex]Now that we have the model, we can find y when x = 10 (2010):[tex]y=21.7(1.0238)^{10}[/tex]First raise 1.0238 to the 10th power to gety = 21.7(1.266097) andy = 27.47