Since populations are continuously changing based on their immediate previous state, the
Pert formula can be used to quickly estimate near future growth (or decay). Assume that the
1990 census for a city of 800,000 reveals an annual increase of 2.3%, estimate the population
for the year 2000.
How long will it take for the population to reach 1 million? You must show your algebraic
work
A=Pe^rt=800,000e^(.023*10)=1,006,880 in year 2000
How long to one million?
1,000,000=800,000e^(.023t)
1.25=e^(.023t) take ln of each side
.233=.023t
t=9.7 years
Never heard A=Pe^rt referred to as the "Pert" formula, but it makes perfect sense.
I guess I am never too old to learn.
To estimate the population for the year 2000 using the Pert formula, we first need to calculate the growth rate over the period from 1990 to 2000.
The annual increase of 2.3% can be converted to a decimal by dividing it by 100: 2.3% = 0.023.
Next, we use the Pert formula to estimate the population at a particular year using the formula:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at year t,
P0 is the starting population (in this case, the population in 1990 = 800,000),
r is the annual growth rate (0.023),
t is the number of years.
To estimate the population in the year 2000, t will be 10 years (2000 - 1990 = 10).
Substituting the values into the formula:
P(2000) = 800,000 * (1 + 0.023)^10
Now we can calculate P(2000):
P(2000) = 800,000 * (1.023)^10
= 800,000 * 1.267
≈ 1,013,600
Therefore, the estimated population in the year 2000 is approximately 1,013,600.
To calculate how long it will take for the population to reach 1 million, we need to find the value of t that satisfies the equation:
1,000,000 = 800,000 * (1 + 0.023)^t
To solve for t, we need to isolate it. Divide both sides of the equation by 800,000:
1,000,000 / 800,000 = (1 + 0.023)^t
1.25 = (1.023)^t
To isolate t, we can take the logarithm (base 1.023) of both sides:
log(1.25) = log[(1.023)^t]
Using logarithm properties, we can rewrite this equation as:
t * log(1.023) = log(1.25)
Now divide both sides of the equation by log(1.023) to solve for t:
t = log(1.25) / log(1.023)
Using a calculator, we can calculate:
t ≈ 16.93
Therefore, it will take approximately 16.93 years for the population to reach 1 million.