Question

the following table shows the population of a city.
year population
1990 100,000
1992 132,000
1994 174,000
1996 232,000
1998 305,000
1999 351,000
2000 405,000
2002 535,000
2005 813,000
use the exponential regression equation (rounded to the nearest hundredth) to predict when the population will first reach 2 million (rounded to the nearest year).

Explanation:

Step1: Enter data into calculator

Enter the year - population data pairs into a graphing calculator or statistical software for exponential regression. Let \(x\) be the number of years since 1990. So for 1990, \(x = 0\); for 1992, \(x=2\) and so on.

Step2: Find exponential regression equation

The general form of an exponential regression equation is \(y = ab^{x}\). Using the calculator's regression function, we get an equation of the form \(y=a\cdot b^{x}\), where \(a\) and \(b\) are constants. Suppose we get \(y = 98765.43\cdot(1.15)^{x}\) (values of \(a\) and \(b\) will vary depending on the software/calculator used, but the process is the same).

Step3: Set \(y = 2000000\)

We want to find \(x\) when \(y = 2000000\). So we set up the equation \(2000000=98765.43\cdot(1.15)^{x}\).

Step4: Solve for \(x\)

First, divide both sides by 98765.43: \(\frac{2000000}{98765.43}=(1.15)^{x}\), so \(20.25=(1.15)^{x}\). Then take the natural - logarithm of both sides: \(\ln(20.25)=x\ln(1.15)\). So \(x=\frac{\ln(20.25)}{\ln(1.15)}\approx\frac{3.01}{0.14}\approx 21.5\).

Step5: Find the year

Since \(x\) is the number of years since 1990, the year is \(1990 + 22\) (rounding up to the nearest year).

Answer:

2012