P(t) = . Predicted population in the year 2000 =million people. Doubling time = years.

Question

The population of a region is growing exponentially. There were 40 million people in 1980 (when ) and 55 million people in 1990. Find an exponential model for the population (in millions of people) at any time , in years after 1980.=What population do you predict for the year 2000?Predicted population in the year 2000 = million people.What is the doubling time?Doubling time = years.

Answer 1

P( new population) = P (old population) e^kt

Where k = continuous growth rate and

t = time

So, the population being in millions,

55 = 40e^10k

55/40 = e^10k

1.375 = e^10k

ln (1.375) = ln e^10k (ln = natural log, lne =1)

ln (1.375) = 10k lne (log rules)

0.3184537311 = 10k

0.03184537311 = k

So the anytime growth formula would be

P_n = P_oe0.03184537311*t

Time for doubling the population:

2 = e^kt Take the natural log on both sides and solve.

ln 2 = kt ln e

ln 2 =0.03184537311* t ln e

0.6931471806/ 0.03184537311 = t

t= 21.77 years

Answer 2

The population is growing exponentially, and so we have a formula associated with exponential growth over time. We must know at time 0, the initial starting point, which is the year 1980, and 40 million people for this problem. We are also given that over a period of 10 years, the population grows to a total of 55 million in the region. The formula is x (at time t) = x (at time 0 - initial) times (1 + r)^t , where t is in years, and r is a fractional number between 0 and 1, usually given in decimal form.

First, we must find the rate, r, given the information and plug into the formula:

55 = 40(1+r)^10

divide each side by 40 => 55/40 = (1+r)^10

take the 10th root of both sides => (55/40)^(1/10) = 1 + r

subtract 1 from both sides => (55/40)^(1/10) - 1 = r

now we can easily put this in our calculator to solve for r: r = 0.0323578627

To answer the first question," how many millions of people are in this region in the year 2000?," we use the formula again using the rate found, and solving for x at time t = 20 years (20 years from the year 1980 is the year 2000). => x = 40(1.0323578627)^20

so, x = 75.625 million people in the region at the time of year 2000.

For a doubling of the population since the initial time in 1980, we want to find time t (the time it takes to double the population since 1980) - so, going from 40 million to 80 million, we again can use the formula:

80 = 40(1.0323578627)^t

divide both sides by 40 => 80/40 = (1.0323578627)^t

simplify => 2 = (1.0323578627)^t

take the log base 1.0323578627 on both sides => log (base 1.0323578627) of 2 = t

use the change of bases to put into log base 10 for simpler calculators to calculate:

log(2)/log(1.0323578627) = t

Use your simple scientific calculator to find t => t = 21.766 years So, between 2001 and 2002, the population of the region will reach 80 million people. :-)