The population of a community is known to increase at a rate

The population of a community is known to increase at a rate proportional to the number of people present at time t. If an initial population P_0 has doubled in 5 years, how long will it take to triple? To quadruple? 2. Suppose it is known that the population of the community in Problem 1 is 10,000 after 3 years. What was the initial population P_0? What will be the population in 10 years? How fast is the population growing at t = 10?

Solution

1. The initial population P₀ has doubled in 5 years. Let at time t, the size of the population is P. Given that the rate of increase in population is proportional to the number of people present at time t.

dP/dt P

Or, dP/dt = kP where k is the constant of proportionality, k > 0

Or, dP/P = kdt

Integrating we get, ln P = kt + C where C is an arbitrary constant

At t = 0, P = P₀ implies ln P₀ = C

Then, ln P = kt + ln P₀

ln (P/P₀) = kt     ----------------------------------------------- (1)

At t = 5, P = 2P₀, then from equation (1) we get, ln 2 = 5k

So, k = (ln 2)/5

To find time t when the population will be tripled

Let at t = T, P = 3P₀, from equation (1) we get, ln 3 = T(ln 2)/5

T = 5(ln 3)/(ln 2) years 7.92 years

It will take 7.92 years (approximately 8 years) to triple the population. (Answer)

To find time t when the population will be quadrupled

Let at time T’, P = 4P₀, from equation (1) we get, ln 4 = T’(ln 2)/5

T = 5(ln 4)/(ln 2) years = 10 years                                         [As, ln 4 = 2 ln 2, (ln 4)/(ln 2) = (2 ln 2)/(ln 2) = 2]

It will take 10 years to quadruple the population. (Answer)

2. Given at t = 3, P = 10,000. From equation (1) we get ln (10000) = 3k + ln P₀

Or, ln (10⁴) = 3k + ln P₀

Or, 4(ln 10) = 3k + ln P₀

Or, ln P₀ = 4(ln 10) – 3(ln 2)/5

Or, P₀ = exp {4(ln 10) – 3(ln 2)/5} 6598

Hence, the initial population was 6598. (Answer)

To find the size of the population P at t = 10:

Substitute, P₀ = 6598, t = 10 in the equation ln P = kt + ln P₀

ln P = 10(ln 2)/5 + 8.79

Or, ln P = 2(ln 2) + 8.79

Or, ln P = 10.18

Or, P = exp(10.18) 26390

The size of the population after 10 years is 26390. (Answer)

Population growth rate at t = 10, is (26390 - 6598)/6598 x 100 = 300% (Answer)