To determine when the population of the small city reached 17188 people, we start by analyzing the given equation:
[tex]\[ P = 124t + 15700 \][/tex]
Here, [tex]\(P\)[/tex] represents the total population, [tex]\(t\)[/tex] represents the number of years after 1976, and we are given that the population reaches 17188 people. We need to solve for [tex]\( t \)[/tex].
1. Identify the known values:
[tex]\[
P = 17188, \quad 124 \text{ (as the coefficient of } t\text{)}, \quad 15700 \text{ (initial population)}
\][/tex]
2. Set up the equation with the known values:
[tex]\[
17188 = 124t + 15700
\][/tex]
3. Isolate the term containing [tex]\( t \)[/tex]:
[tex]\[
17188 - 15700 = 124t
\][/tex]
4. Simplify the left-hand side:
[tex]\[
1488 = 124t
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{1488}{124} = 12
\][/tex]
6. Determine the year when the population reached 17188:
Since [tex]\( t \)[/tex] represents the number of years after 1976, we add [tex]\( t \)[/tex] to 1976:
[tex]\[
1976 + 12 = 1988
\][/tex]
Therefore, the population reached a total of 17188 people in the year [tex]\(\boxed{1988}\)[/tex].
