Question

A country's population in 1995 was 151 million. in 2001 it was a 156 million. estimate the population in 2003 using the exponential growth formula. round to the nearest million

Answer 1

Answer:

Step-by-step explanation:

Let's set some rules here first regarding how we are going to number our years.  Let's say that 1995 is the 0 year.  1996 is year 1; 1997 is year 2; 1998 is year 3; etc.  That means that 2001 is year 6.  The coordinates that result from this substitution are

(0, 151M) and (6, 156M) (We will lose the "M" til the end from here on.)

The standard form of an exponential growth equation is

$y=a(b)^x$

where x and y are the coordinates from our points, a is the intial value, and b is the growth rate.  We need these x and y values from our 2 coordinates to solve for a and b.  Making our first replacement with our first set of coordinates where x = 0 and y = 151:

$151=a(b)^0$

b raised to the power of 0 (ANYTHING raised to the power of 0, actually) is 1, therefore:

a = 151

Now we will use that value of a along with the x and y from the second coordinate pair to solve for b:

$156=151(b)^6$

We will divide both sides by 151 to get a decimal:

$1.033112583=b^6$

In order to solve for b we have to get rid of that 6th power.  We "undo" the 6th power by taking the 6th root of it.  Because this is an equation, we than have to take the 6th root of the other side too, giving us

$\sqrt[6]{1.033112583}=b$

That means that

b = 1.005444127

With these values now, the model we will use to find the population in 2003 (which is year 8) looks like this:

$y=151(1.005444127)^x$

Now we will sub in an 8 for x (the year) and solve for y (the population):

$y=151(1.005444127)^8$

Raise 1.005444127 to the 8th power first to get

y = 151(1.044391992)

and then multiply 151 by that decimal to get

y = 157.7 million or, rounded to the nearest million,

y = 158 million