in 2013 the population was 5400.
that's a loss of 300 in 3 years which becomes an average loss of 300 / 3 = 100 per year.
if this trend continues, the population in 2016 is predicted to be 5400 - (2016 - 2013) * 100 = 5400 - 3 * 100 = 5400 - 300 = 5100.
this assumes a straight line loss which is a constant amount per year.
the equation for the straight line loss would be y = 5700 - 100 * x, where x is the number of years since 2010.
this can be graphed and looks like this.
when x = 3, the year is 2010 + 3 = 2013.
when x = 6,the year is 2010 + 6 = 2016.
the above formuls is used if the drop in the population is at a constant amount per year.
if the drop in the population is at a constant rate per year, then the formula is different.
to calculate the average rate of loss, use the following formula.
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period (assumed to be years in this problem).
n is the number of time periods (assumed to be years in this problem.
since p = 5700 and f = 5400, and n = 3 (from 2010 to 2013), the formula becomes:
5400 = 5700 * (1 + r) ^ 3
divide both sides of this equation by 5700 to get 5400 / 5700 = (1 + r) ^ 3
take the third root of both sides of this equation to get (5400 / 5700) ^ (1/3) = 1 + r.
subtract 1 from both sides of this equation to get (5400 / 5700) ^ (1/3) - 1 = r
solve for r to get r = -.0178609748.
that's your average interest rate per year.
confirm by using that rate in the formula of f = p * (1 + r) ^ 3
when p = 5700 and r =-.0178609748, the formula becomes f = 5700 * (1 - .0178609748) ^ 3 = 5400, which is correct, so the interest rate per year is good.
to find the population in s016, use the formula of f = p * (1 + r) ^ n, where:
p = 5700
r =-.0178609748
n = (2016 - 2010) = 6
to get f = 5700 * (1 - .0178609748) ^ 6 = 5115.789474 which rounds to 5116.
this formula can be graphed and looks like this.
the difference between the methods is not so pronounced in only 6 years.
if you go out more years, the difference will be more pronounced.
going out 20 years, or even 30 years, you can see the difference in the following graph that shows both methods on the same graph.
the red line is the constant amount per year loss.
formula is y = 5700 - 100 * x.
the blue line is the constant rate per year loss.
formula is y = 5700 * (1 - .0178609748) ^ x.
the values for the values of x indicated are:
year x red line blue line blue line minus red line 2010 0 5700 5700 0 2013 3 5400 5400 0 2016 6 5100 5116 16 2030 20 3700 3975 275 2040 30 2700 3319 619