The population of a small town was 500 in 2018 and is growing ...

Question

The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says the price of a toy was $24 in the year 2000, and every year the price grows by $2. Create the price function PFX where X is the number of years after the year 2000. Draw the graph of PF X and calculate the price of the toy in 2013. So, we need to answer 3 parts of this question. So, the first part we're going to look at is creating our price function PFX. So we're gonna have to use the information in our problem to create this function. So, PF X is going to be equal to. And if we look at the very first sentence that says that the price of the toy was $24 in the year 2000, and every year the price grows by $2. So, for every year, the price grows $2. That means we're going to have two, multiplied by the number of years after 2000, which is X. So we'll have two multiplied by X. However, when X is zero, we're gonna be in the year 2000, and the price of the toy in the year 2000 is $24. So I'll add to this $24. So P of X is equal to 2 multiplied by X plus 24. And so that is one part of this question answered. So that is our price function. We now need to draw this price function on a graph. And since we have a linear function here, since 2 x + 24 is a linear function, we really just need to determine two points in order to draw our line on our graph. So we're going to choose the two points when X is equal to 0 and X is equal to 2. So when X is equal to 0, we will have p of 0. is equal to 2 multiplied by 0, plus 24, and this is going to be equal to 24, which was the initial price of our toy. Then we'll be looking at our price when X is equal to 2, so we'll have PF 2. And that's going to be equal to 2, multiplied by 2. Plus 24, and this is equal to 28. So now we need to plot this on a graph. So, on a graph, we will have P on the Y axis and X on the X-axis. And so we're going to label our two points on this graph. So, the first point occurred when X is equal to 0 and P of X is equal to 24. We'll place a point at 0.24, and then the next point that we calculated was when X was equal to 2 and P of X was equal to 28. And so we'll place another point at 2.28. And so now we will just draw a line connecting those two and extend that line out. Across the rest of the graph. And so, in doing so, we'd get a graph that looks similar to what is seen on the screen. We have a linear line that has a slope of 2, and begins at a Y value of 24, when X is equal to 0. So this is the next part of the question answered. Now for the final part of the question, we need to calculate the price of the toy in 2013. So, that means that 2013 occurs 13 years after the year 2000, so we're gonna be looking for PF 13. And this is going to be equal to 2, multiplied by 13. Plus 24. And when we evaluate that expression, this is going to be equal to 50. So that would be $50. And so that is the Answer to the last part of the question. So now we have answered actually all parts for this question. So, I hope that this explanation has been useful, and I'll see you in the next video.

The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.

Key Concepts

Linear Functions

Slope and Intercept

Graphing Linear Functions

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