Question

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​

Answer 1

Answer:

approximately 27.5 million

Step-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

$y=a(b)^x$

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):

$21.7=a(b)^0$

andything rised to the power of 0 = 1, so b raised to 0 = 1:

21.7 = a(1) so

a = 21.7

Now we use coordinate pair (6, 25) in equation 2, subbing in our value for a also:

$25=21.7(b)^6$

Divide both sides by 21.7 to get

$1.152073733=b^6$

We "undo" that power of 6 by taking the 6th root of both sides:

$(1.152073733)^{\frac{1}{6}} =(b^6)^{\frac{1}{6}}$

That gives you that

b = 1.0238 (rounded).

Now that we have a and b, we can write the model for this situation:

$y=21.7(1.0238)^x$

Now that we have the model, we can find y when x = 10 (2010):

$y=21.7(1.0238)^{10}$

First raise 1.0238 to the 10th power to get

y = 21.7(1.266097) and

y = 27.47