Question

The population of a town with a 2016 population of 66,000 grows at a rate of 2.5% per year a. Find the rate constant k and use it to devise an exponential growth function that fits the given data b. In what year will the population reach 176.000? Book a. Find the rate constant k k= (Type an exact answer) tents ccess Library Resources

Answer 1

Answer: a) k= 0.025

The exponential growth function in time t is given by :-

`P=66000e^(0.025t)`,

b) In year 2033 the population will reach to 176,000.

Explanation:

The exponential growth function in time t is given by :-

`P=P_0e^(kt)`, where k is the rate of growth ,
`P_0` is the initial population.

Given : In 2016 , the initial population of town =
`66,000`

The rate of growth per year=
`k=2.5\%`

Which can be written as

`k=0.025`

Let t be the number of years since 2016 to take population reach 176,000.

Then , the required equation will be :-

`176000=66000e^(0.025t)\\\\\Rightarrow\ e^(0.025t)=(176)/(66) \\\\\Rightarrow\ e^(0.025t)=2.67`

Taking log on both sides , we get

`0.025t=\log(2.67)\\\\\Rightarrow\ 0.025t=0.426511261365\\\\\Rightarrow\ t=17.0604504\approx17`

Thus it will take 17 years since 2016 to reach population 176,000.

Hence, In year 2033 the population will reach to 176,000.