Exponential Models

Question

The population of a certain city was 120,000 in 2014, and the observed doubling time for the population is 19 years.

a) Find an exponential model n(t) = n02^t/a for the population t years after 2014.

b) Find the exponential model n(t) = n0e^rt for the population t years after 2014.

c) show a graph

d) estimate how long it takes the population to reach 500,000

Answer 1

a) n(t)=n₀2^(t/a) now at time t=0, n=n₀ but at t=19 years, n=2n₀

so

n(t=19 years) = n₀(2)^(19/a) which must be equal at that time to 2n₀

so n₀(2)^(19/a) = 2n₀ so 2^19/a = 2 so 19/a = 1 so a = 19 years

and n(t) = n₀ 2^t/19

b) n(t) = n₀e^(rt) to find what r, the coefficient of t in the exponent is,

n(t=19 years)= n₀e^{r(19 years)} =2n₀

so e^{r(19 years)} = 2

we take the NATURAL logarithm of both sides (that the logarithm for base e)

r(19) = ln(2)

r = ln(2)/19 = 0.0365 year^-1

so n(t) = n₀e^(rt) = n₀e^,0365t

c) Sorry I don't have the graph capability.

d) what time does it take to get for the population to reach 500,000?

You can use either of the two expressions; I'll use the one from a)

n(t) = n₀ 2^t/19= 500000

n₀= 120000

so (120000) 2^t/19= 500000

2^t/19 = (500000/120000)=4.17

take the natural log of both sides

ln(2^t/19) = ln(4.17)

Note that ln(a^b) = b ln(a) so

(t/19)ln(2) = ln(4.17)

t= 19 ln(4.17)/ln(2) ≅39.1 years (as a check notice that since the population doubles every 19 years, it will quadruple in twice that time 38 years, at which time the population with be 480000, so it makes sense it would reach 500000 a relatively short time after that.