Question

The population of a town in 1921 is 2900 people the towns population decrease linearly mean in 1928 the population of 2700. Find the formula for P,the towns population in terms of t, the number of years since 1920

Answer 1

We know that

• The number of years starts in 1920, this is the initial condition.

,

• The population of a town was 2900 people.

,

• In 1928 the population was 2700.

First, we have to transform the given information into math expression. From the given data we deduct two points (0, 2900) and (8, 2700). The first point refers to the population at the beginning of the study (1920), the second point refers to the population in 1928 (eight years after the beginning of the study.

To find the function, we have to find the slope first. The slope formula is

`m=(y_2-y_1)/(x_2-x_1)`

Where the coordinates are

`\begin{gathered} x_1=0 \\ x_2=8 \\ y_1=2900 \\ y_2=2700 \end{gathered}`

Then, we use the values above to find the slope.

`m=(2700-2900)/(8-0)=(-200)/(8)=-25`

The slope is -25.

At this point, we know the slope and the y-intercept. So, we can express the formula for P in slope-intercept form.

`y=mx+b`

Where m = -25 and b = 2900 (remember that the y-intercept is the initial condition of the problem).

Therefore, the formula for the population P is

`P(t)=-25t+2900`

In this case, x is t.