It is estimated that t years from now the population of a ce

It is estimated that t years from now, the population of a certain suburban community will be P(t) = 23 - 6/t + 1 thousand people. (a) What will the population of the community be seven years from now? people (b) By how much will the population increase during the seventh year? (Round your answer to the nearest integer.) people (c) What will happen to the size of the population in the \"long run\"? The population will tend to people in the long run.

Solution

To find the population in 7 years, we plug t = 7 into our function to get

P (7) = 23 - 6/(7-1) = 22

b) To find out how much the population increases during the 7tg year, observe that the first year goes from t = 0 to t = 1, the second year from t = 1 to t = 2, etc.
So the 3rd year then goes from t = 2 to t = 3. and do on We then need to find P(6) and subtract from P(7) to find the increase during that time period. We have

P (6) = 23 - (6/5 ) = 109/5

P (7) = 23 - (6/6) = 22

P (7)-p (6) = 109/5 - 22 = -1/5

C) To see what happens as t gets larger and larger, observe that

6/(t-1) > 0
will get smaller as t gets larger, until it ultimately ends up being zero. So we are subtracting smaller and smaller amounts from 23. So the population will increase at a diminishing rate, slowly approaching 23,000 inhabitants in the long
term.