The population of a southern city follows the exponential law. If the population doubled in size over 15 months and then current population is 40,000, what will the
population be 4 years from now?
pop = 40000(2)^(t/15) where t is in months
So replace t with 48
that is, pop = 40000(2)^(48/15)
= ....
To find the population 4 years from now, we first need to determine the rate at which the population is growing. Since the population follows the exponential law, we can use the formula:
P(t) = P₀ * e^(rt)
Where:
P(t) is the population after time t
P₀ is the initial population
e is Euler's number (approximately 2.71828)
r is the growth rate
t is the time in the same units as the growth rate
We know that the current population P₀ is 40,000. We also know that the population doubled in size over 15 months. This means that after 15 months, the population became 2 * 40,000 = 80,000.
We can use this information to find the growth rate r:
80,000 = 40,000 * e^(15r)
Dividing both sides of the equation by 40,000, we get:
2 = e^(15r)
To solve for r, we take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(15r))
Using the property of logarithms that ln(e^x) = x, we can simplify the equation to:
ln(2) = 15r
Now we can solve for r:
r = ln(2) / 15
Using a calculator, we find that r ≈ 0.04620.
Now, we can use the formula to find the population 4 years from now (t = 4 years):
P(t) = P₀ * e^(rt)
P(4) = 40,000 * e^(0.04620 * 4)
Using a calculator, we find that P(4) ≈ 54,317.
Therefore, the population of the southern city will be approximately 54,317 four years from now.