4. The population of a small town is modelled by the function

Question

p(t)= 20(4t+3)/2t+5
, where P(t) is the population, in thousands, and t is time, in years, since the start of 1990.
a) Is the population currently increasing or decreasing? Justify your answer.

b) The town will need its own transit system if the population exceeds
50 000. Will the town’s population ever exceed 50 000? Explain.

Answer 1

a little long division shows that
p(t) = 40 - 140/(2t+5)

As t increases, a smaller and smaller amount is subtracted from 40, so for all t>0, p(t) is increasing.

But, no matter how big t gets, p(t) < 40, so it indicates that the population will never exceed 50,000.