Question

The population of a small town in central Florida has shown a linear decline in the years 1985-1993.In 1985 the population was 31800 people. In 1993 it was 24600 people.A) Write a linear equation expressing the population of the town, P, as a function of t, the numberof years since 1985.Answer:B) If the town is still experiencing a linear decline, what will the population be in 1996?

Answer 1

We must represent as a function "P" as a function of "t" years elapsed the decrease in population in a town in Florida.

For this, we have two points that have input values which are the years, and output points which are the population.

With these points and the equation of the line in the form slope-intercept, we can find this function. The equation of the line in the form slope-intercept is:

`y-y_1=\frac{y_2-^{}y_1}{x_2-x_1}(x-x_1)`

Where in this case:

`\begin{gathered} (t_1,p_1)=(1985,31800) \\ (t_2,p_2)=(1993,24600)_{} \end{gathered}`

Now, we replace and solve:

`\begin{gathered} p-31800=(24600-31800)/(1993-1985)(x-1985) \\ p-31800=\frac{-7200}{}(x-1985) \\ p-31800=-900(x-1985) \\ p-31800=-900x(-900\cdot-1985) \\ p-31800=-900x+1786500 \\ p=-900x+1786500+31800 \\ p=-900x+1818300 \end{gathered}`

In conclusion, the linear equation expressing the population of the town is:

`p=-900x+1818300`

On other hand, we need to find the population in 1996 if the linear decline were to continue, for this, we take x = 1996 as the input value.

`\begin{gathered} x=1996 \\ p=-900x+1818300 \\ p=-900\cdot(1996)+1818300 \\ p=-1796400+1818300 \\ p=21900 \end{gathered}`

In conclusion, the population in 1996, if the linear decline were to continue, is 21900.