Final answer:
To find the population of a small town in North Dakota in 1998, we integrate the population change function from 1991 to 1998 and add it to the initial population, resulting in an approximate population of 3179.
Step-by-step explanation:
The student asked about calculating the population in a small town in North Dakota in the year 1998, given that the change in population per year can be modeled by the function r(t) = 36 - 3t, where t=0 corresponds to the year 1991, and the population in 1991 was 3000. To find the population in 1998, we need to integrate the function r(t) from t=0 to t=7 (since 1998 is 7 years after 1991) and then add this value to the initial population.
The integral of r(t) is R(t) = 36t - 1.5t^2 + C. When t=0, the population is 3000, which gives us the value of C (3000 = 36(0) - 1.5(0)^2 + C, hence C=3000). To find the population in 1998, we evaluate R(7) = 36(7) - 1.5(7)^2 + 3000.
R(7) = 252 - 73.5 + 3000 = 3178.5
Therefore, the population in the year 1998 was approximately 3179 (since population is typically rounded to an integer).