Question

In 2012, the population of a city was 5.13 million. The exponential growth rate was 2.64% per year

a) Find the exponential growth function.
b) Estimate the population of the city in 2018.
c) When will the population of the city be 8 million?
d) Find the doubling time.

Answer 1

a. The exponential growth function is:
`[P(t) = 5.13 * e^(0.0264t)]`

b. The population of the city in 2018 is 5.94 million.

c. The population will reach 8 million in about 11.5 years, which is around 2023.

d. The doubling time is:26.2 years.

a) The exponential growth function can be found using the formula:
`[P(t) = P_0 * e^(rt)]`where (P(t)) is the population at time (t), (P_0) is the initial population, (r) is the growth rate, and (t) is the time in years. For the given city with an initial population of 5.13 million and a growth rate of 2.64% per year, the exponential growth function is:
`[P(t) = 5.13 * e^(0.0264t)]`

b) To estimate the population of the city in 2018, we can substitute (t = 6) into the exponential growth function:
`[P(6) = 5.13 * e^(0.0264 * 6) \approx 5.94 \text{ million}]`

c) To find when the population of the city will be 8 million, we need to solve for (t) in the equation:
`[8 = 5.13 * e^(0.0264t)]` This gives:
`[t \approx (\ln(8/5.13))/(0.0264) \approx 11.5 \text{ years}]` So, the population will reach 8 million in about 11.5 years, which is around 2023.

d) The doubling time can be found using the formula:
`[T = (\ln(2))/(r)]`where (T) is the doubling time and (r) is the growth rate. For the given growth rate of 2.64% per year, the doubling time is:
`[T \approx (\ln(2))/(0.0264) \approx 26.2 \text{ years}]`