Brief Calculus.

Question

The population of a small town has shown a linear decline in the years from 2000 to 2020. In the year 2000 the population was 32,000 people. In the year 2010 the population was 27600 people.

a) Write a linear function relating the population of the town as a function of the year.

b) What is the slope? Write a sentence interpreting the meaning of the slope in this scenario.

Answer 1

A linear function is one that fits the format y = mx + b. To complete the function, you need "m" and "b".

"m" is the slope. The slope formula is (y₂ - y₁)/(x₂ - x₁)

You were given the data points(2000, 32000) and (2010, 27600) so the slope is (27600 - 32000)/(2010 - 2000) = (-4400)/10 = -440. That means the population is dropping by 440 people per year.

You now can write the function as y = -440x + b however you still must find "b". To do so, plug in one of the data points as the "x" and "y". I'll use (2000, 32000) so:

32000 = -440(2000) + b

32000 = -880000 + b

880000 + 32000 = b

b = 912000

So the function is y = -440x + 912000

If you want, you can change the variables to match the problem. Instead of the year being "x", you can use "y" for year. And then you can make the population be P(y). That makes the function P(y) = -440y + 912000.