a) To find the exponential growth function, we can use the formula:
P(t) = P0 * e^(rt)
Where:
P(t) is the population at time t
P0 is the initial population
e is the base of the natural logarithm (approximately 2.71828)
r is the growth rate
t is the time in years
In this case, the initial population (P0) is 5.95 million and the growth rate (r) is 2.37% per year. Converting the growth rate to a decimal, we have r = 0.0237.
Therefore, the exponential growth function for this city's population is:
P(t) = 5.95 * e^(0.0237t)
b) To estimate the population of the city in 2018, we need to substitute t = 2018 - 2012 = 6 into the exponential growth function:
P(6) = 5.95 * e^(0.0237 * 6)
Using a calculator or computer program, we can evaluate this expression to find the estimated population in 2018.
c) To find when the population of the city will be 9 million, we need to solve the exponential growth function for t:
9 = 5.95 * e^(0.0237t)
Dividing both sides by 5.95, we get:
1.514286 = e^(0.0237t)
To solve for t, we take the natural logarithm of both sides:
ln(1.514286) = ln(e^(0.0237t))
Using properties of logarithms, we can simplify this equation to:
ln(1.514286) = 0.0237t
Now, divide both sides by 0.0237:
t = ln(1.514286) / 0.0237
Using a calculator or computer program, we can evaluate this expression to find when the population will reach 9 million.
d) The doubling time is the amount of time it takes for the population to double. To find this, we need to solve the exponential growth function for t when P(t) = 2P0:
2P0 = P0 * e^(rt)
Dividing both sides by P0, we get:
2 = e^(rt)
Taking the natural logarithm of both sides, we have:
ln(2) = rt
Now, divide both sides by r:
t = ln(2) / r