Question

The population (in millions) of a certain country can be approximated using the following function; P(x)=40 ( 1.02)^x where x is the number of years after 2000. which of the following calculations will tell you what the population can be expected to reach 200 million?

Answer 1

To find the year when the population can reach 200 million, set up the equation P(x) = 200 and solve for x. The population can be expected to reach 200 million in approximately 22 years after 2000, around the year 2022.

Step-by-step explanation:

To find the year when the population can be expected to reach 200 million, we need to set up the equation P(x) = 200 and solve for x. The equation P(x) = 40 * (1.02)^x represents the population as a function of the number of years after 2000. We substitute 200 for P(x) and solve for x:

200 = 40 * (1.02)^x

Divide both sides of the equation by 40:

5 = (1.02)^x

Take the logarithm of both sides of the equation:

log(5) = log((1.02)^x)

Apply the logarithmic property to move the exponent x down:

log(5) = x * log(1.02)

Divide both sides of the equation by log(1.02):

x = log(5) / log(1.02)

Using a calculator, we find that x is approximately 22.0902. Therefore, the population can be expected to reach 200 million in approximately 22 years after 2000, which is around the year 2022.

Answer 2

ln(5)/ln(1.02)+2000

Step-by-step explanation:

200=40•1.02^x

5=1.02^x

ln(5)=ln(1.02^x)

ln(5)=x•ln(1.02)

ln(5)/ln(1.02)=x

Since X is after the year 2000, you have to add the 2000 back to get the full amount of years, so the answer is ln(5)/ln(1.02)+2000