Question

If the population of a country is reported in the year 1990 as 471.56 million and in the year 2000 as 492.53 million, when will the population reach 550 million (round your answer to the nearest whole number)? 15 years 35 years O 25 years 5 years

Answer 1

Answer:

35 years

Step-by-step explanation:

It will make this problem a whole lot easier if we call the year 1990 our initial year, or year 0; that means that the year 2000 is year 10.  The coordinates then that result from that change are (0, 471.56) and (10, 492.53).  Population growth is always exponential, and the formula for that, in terms of x and y, where x is the year and y is the population in millions, is

$y=a(b)^x$

a is the initial population at year 0 and b is the growth rate.  If the growth rate is a decimal (less than 1) what we have is decay as opposed to growth.  In order to determine when (x) the population (y) is 492.53, we have to find the model by solving for a and b using the coordinates.  Once we find this model, we will use it to solve for x by filling in the y of 550 million.  So here we go.

Using the first coordinate, we know that at year 0 the population is 471.56.  So that is our initial population, a in our formula.  Using that value of a and plugging it in to the formula with the other coordinate gives us

$492.53=471.56(b)^{10}$

Now we have to solve for b.  Begin by dividing both sides by 471.56 to get

$1.044469421=b^{10}$

"Undo" that 10th power by taking the 10th root of both sides which gives you that b = 1.004360381

Now we have our model:

$y=471.56(1.00436)^x$

We will plug in a y value of 550 and solve for x using natural logs.

$550=471.56(1.004360381)^x$

Begin by dividing both sides by 471.56 to get

$1.166341505=(1.004360381)^x$

In order to solve for x we have to pull it down from the position in which it is currently sitting, which is an exponent.  Taking the natural log of both sides will solve that problem for us:

$ln(1.166341505)=ln(1.004360381)^x$

The power rule for natural logs tells us we can pull the exponent down in front giving us:

$ln(1.166341505)=x*ln(1.004360381)$

Now we will divide both sides by ln(1.004360381) to isolate the x.  Doing that and at the same time finding those values on our calculator gives us:

$\frac{.153871931}{.0043505227}=x$ so

x = 35