Average Rate of Change - Population

[Calc12]

The population of a town is modelled by P(t) = 6t^2 + 110t + 3000, where P is the population and t is the number of years since 1990.


a) find the average rate of change in population between 1995 - 2005


b) Estimate the rate at which the population is chaning in 2005.



* I am currently taking Calculus and Vectors for the first time through corrospondance, with no textbook or other material support. I would really appreciate any help.

Thank you very much in advance.

 

[soroban]

Hello, Calc12!

The population of a town is modelled by: .\(\displaystyle P(t) \:=\: 6t^2 + 110t + 3000,\)
where \(\displaystyle P\) is the population and \(\displaystyle t\) is the number of years since 1990.

a) Find the average rate of change in population between 1995 - 2005

\(\displaystyle \text{In 1995 }(t = 5)\text{, the population was: }\(5) \:=\:6(5^2) +110(5) + 3000 \:=\:3700\)

\(\displaystyle \text{In 2005 }(t = 15)\text{, the population was }\(15) \:=\:6(15^2) + 110(15) + 3000 \:=\:6000\)

\(\displaystyle \text{The population increased from 3700 to 6000 over 10 years.}\)

\(\displaystyle \text{The average rate of change is: }\:\frac{2300}{10} \:=\:+230\)

\(\displaystyle \text{The population was increasing at an average of 230 people per year.}\)

b) Estimate the rate at which the population is changing in 2005.

\(\displaystyle \text{The derivative provides the instantaneous rate of change.}\)

\(\displaystyle \text{We have: }\;P'(t) \:=\:12t + 110\)

\(\displaystyle \text{Therefore: }\'(15) \:=\:12(15) + 110 \:=\:290\)

 

[Calc12]

Thank you