Respuesta :
ANSWER
The formula is P(t) = 280t + 4220
In 2006 the population will be 8700
EXPLANATION
If the population changes linearly, we're looking for a formula like:
[tex]P(t)=mt+P_0[/tex]P0 is the initial population, in 1990. m is the slope and t is the time in years since 1990.
We have two points (t, P(t)):
• (1, 4500) --> 1 year after 1990 the population was 4500
,• (6, 5900) --> 6 years after 1990 the population was 5900.
With this information we can find the slope m:
[tex]m=\frac{\Delta P}{\Delta t}=\frac{5900-4500}{6-1}=\frac{1400}{5}=280[/tex]The slope is 280. For now, the formula is:
[tex]P(t)=280t+P_0[/tex]To find the y-intercept P0, we have to use one of the points. Using the first point (1, 4500) replace P(t) = 4500 and t = 1 and solve for P0:
[tex]\begin{gathered} 4500=280+P_0 \\ P_0=4500-280 \\ P_0=4220 \end{gathered}[/tex]The formula is:
[tex]P(t)=280t+4220[/tex]To find the population in 2006 we have to know how many years after 1990 is 2006:
[tex]2006-1990=16[/tex]We have to replace t = 16 in our formula:
[tex]\begin{gathered} P(16)=280\cdot16+4220 \\ P(16)=8700 \end{gathered}[/tex]