The given table shows data for federal spending and unemployment rates, per day, for six states in a certain country. A political consultant uses the data in this table to analyze the relationship between the federal spending of each state and the total number of unemployed people. He calculates that the line of best fit that relates to the total number of unemployed people, 𝑢, to the total federal spending in dollars, 𝑥, is given by 𝑢 equals 𝑘𝑥 plus 18000, where 𝑘 is a positive constant. If the data for state C matches this model perfectly, what is the value of 𝑘?
So in this question, we’re given a lot of information. But there are two key things that we need to look at. One is our model. And that model is that 𝑢 is equal to 𝑘𝑥 plus 18000 and the other is the information relating to state C. And that information tells us that the population of state C is 3900000. The federal spending for state C is 280 multiplied by 10000 dollars and the unemployment rate is 22 percent. So if we have our model 𝑢 equals 𝑘𝑥 plus 18000, where 𝑢 is the number unemployed and 𝑥 is our federal spending, what we’re gonna do is rearrange this to find 𝑘 and we’re gonna do that using the information that we’ve got in the table.
So the first thing we need to work out is 𝑢, so the number unemployed. And the number unemployed is 22 percent of the total population of state C. So it’s 22 percent of 3900000. So to work out 𝑢, what we do is we multiply 0.22, our decimal multiplier, by 3900000. And the reason that we did multiplying by our 0.22 is because 22 percent means 22 out of 100 or 22 divided by 100. And 22 divided by 100 is 0.22. So that’s our decimal multiplier. So therefore, when we calculate this, we get a number unemployed of 858000. So we’ve now found our 𝑢.
So now what we need to do is work out our 𝑥. And our 𝑥 is our federal spending. And to work this out, what we need to do is multiply 280 by 10000 dollars because we’re told that that’s how the column in the table works. So when we multiply 280 by 10000, we get 2800000 dollars. So that is our federal spending. And we can see that one way of thinking about this is when we multiply by 10000 is like adding four zeros on. In real terms, what we’re doing is we’re moving each digit four place values to the left.
So now that we’ve got our 𝑢 which is 858000 and our 𝑥 which is 2800000, what we can do is substitute them into our model to find our constant 𝑘. So when we do that, we get 858000, our 𝑢, is equal to 2800000𝑘 plus 18000. So then, what we need to do is subtract 18000 from each side of the equation. And when we do that, we get 840000 is equal to 2800000𝑘. So now if we want to find single 𝑘 because we’re gonna try and find our constant, what we’re gonna do is divide each side of the equation by 2800000. And this gives us an answer of 0.3.
So we can say that if the model for the line of best fit that relates the total number of unemployed people 𝑢 to the total federal spending in dollars 𝑥 is given by 𝑢 equals 𝑘𝑥 plus 18000, where 𝑘 is a positive constant, if the data of the state C matches the model perfectly, the value of 𝑘 is 0.3.