Hi please help me asap with my algebra homework please

Question

Using an exponential model for the data, estimate when the population will exceed 100 million.

A.) The population will exceed 100 million in 2022.

B.) The population will exceed 100 million in 2011.

C.) The population will exceed 100 million in 2017.

D.) The population will exceed 100 million in 2008.

Answer 1

Hi June!

To solve most exponential modelling problems, we can use the equation A=Pe^rt, where A is the current amount, P is the initial amount, r is the percentage growth rate, and t is the amount of time that has passed between P and A. e is Euler's number, equal to approximately 2.72 (this, like pi, is always constant).

Because your problem gives four data points to work with, we can pick two of those, plug the relevant numbers into the equation, and solve for the growth rate r. Using the first and last lines of data, the initial amount is P=53 million, the "current" amount (as of 2006) is 67 million, and the amount of time passed is 2006 - 2000 = 6 years. So, 67 = 53e^r*6. Solving the equation for r, we divide by 53 {1.26 = e^r*6}, take the natural log of both sides {0.2344 = r*6}, and divide by 6 {r = 0.0391}. So the annual growth rate is 0.0391 or 3.91%.

Now that we know r, we can return to the equation and solve for the time when the "current" population will be 100 million. 100 = 53e^0.0391*t. Solving for time using the same steps as before, we get 16.23, so the population will first reach 100 million 16.23 years after the year 2000, which, rounding up, means this will occur in 2017.