A growing City had a population of 1,000,000 in 2006. In 2015, the population was 2,000,000 assume exponential growth? In what year will the population reach 3 million?

Answer 1

The equation for exponential growth is new population = old population ( 1+%rate/100) ^ years.

2015 - 2006 is 9 years. The equation then is 2,000,000 = 1,000,000( 1 + x/100) ^ 9. Dividing both sides of the equation by 1,000,000 the equation becomes 2 = (1 + 1/100) ^9. Trying the numbers 1-9 for x I find x = 8 makes the equation close to true. 2 = 1.08 ^ 9 because 1.08 ^ 9 = 1.999. Now I put in the numbers and find the exponent. 3,000,000 = (2,000,000 )1.08 ^ y. Y is the years since 2015. I divide both sides of this equation by 2,000,000 to have 1.5 = 1.08^y. By guessing 1-8, I find the answer between 5 and 6. Some more trials give me 5.3 as the exponent. Thus the answer is 2015 + 5.3 years or one quarter into 2020. This answer is by intelligent guessing I could have used logarithms but I don'r know if you have had those yet.

Answer 2

Exponential equations have the form, Y = aX^2 + b. Wee need to find a and b to answer the question. Let's assign X to be time and Y to be population. I'll assume all populations are in Millions to simplify the calculations. One point on our curve will be (0, 1) 2006 will be year zero for us.All this says is that at year zero (2006) the population will be 1 Million. Let's put this point into our exponential equation.

1 = a*0 + b, solving for b, we get b = 1.

the next point we know is (9,2) in other words, at time = 9 (2015) the population is 2 Million.

Let's write our equation with this information.

2 = a*9^2 + 1, solving for a we get a = 1/81.

Now we have a & b!

So our equation looks like this. Y = 1/81x^2 + 1

Let's finish!

Let's write our equation with the question numbers in it.

3 = 1/81x^2 + 1

solving this for X, we get X = 12.7 years.

So the final answer is The population will reach 3 Million in 2018 {2006 + 12.7}!

Hope this helps!