# The population of a cit grows at a rate of 5% each year. The population in 1990 was 400,000. What would be the predicted current population? In what year would we predict the population to reach 1,000,000?

Apr 29, 2016

11 October 2008.

#### Explanation:

Rate of growth for n years is $P {\left(1 + \frac{5}{100}\right)}^{n}$

The starting value of $P = 400 000$, on 1 January 1990.

So we have $400000 {\left(1 + \frac{5}{100}\right)}^{n}$

So we need to determine $n$ for

$400000 {\left(1 + \frac{5}{100}\right)}^{n} = 1000000$

Divide both sides by $400000$

${\left(1 + \frac{5}{100}\right)}^{n} = \frac{5}{2}$

Taking logs

$n \ln \left(\frac{105}{100}\right) = \ln \left(\frac{5}{2}\right)$

$n = \ln \frac{2.5}{\ln} 1.05$

$n = 18.780$ years progression to 3 decimal places

So the year will be $1990 + 18.780 = 2008.78$

The population reaches 1 million by 11 October 2008.