An electrician charges a call-out fee and an hourly labor charge. The graph represents what the electrician charges in dollars for jobs of different durations. What is the electrician’s call-out fee? The second part of the question is, what is the hourly labor charge? And finally, let 𝑦 be the cost in dollars for a job that takes 𝑥 hours. Write an equation for 𝑦 in terms of 𝑥.
So, for the first part of the question, we want to work out the call-out fee. But what is a call-out fee? Well, the call-out fee is the amount that is chargeable regardless of how many hours the electrician works. So therefore, what we can say is it’s the a charge that an electrician would give at zero hours. So, what we can do is have a look on the 𝑥-axis, which is our time axis, and go to zero hours. And we can see that the charge is 40 or 40 dollars. So therefore, we can say that the electrician’s call-out fee is 40 dollars. So It’s also worth noting that this’s, in fact, the 𝑦-intercept, which can also be known as 𝑏 when we use 𝑦 equals 𝑚𝑥 plus 𝑏. So, we’re gonna have a look at that a bit because that will help us with the third part of the question.
So great, we’ve worked out the first part. We know the electrician’s call-out fee. Now what we want to do is work out the hourly labor charge. Well, to work out the hourly labor charge, what we’re gonna do is take a look at the first and second hours, and how much it costs after one hour’s work and after two hours’ work. Well, we can see that after one hour, it’s 100 dollars. And after two hours, it’s 160 dollars. And we know that it’s the same if we moved up to three hours, four hours, five hours. It’s gonna be the same difference per hour because we’ve got a linear relationship or a straight line. And all we need to do is subtract 100 from 160 cause that will show the difference between the two hours and the rates charged for them. And that is 60 dollars.
So therefore, we can say that the hourly labor charge is 60 dollars. And we can check that by looking at hour three. Well, if we get after three hours, the charge is 220. Well, 160 add 60 is 220. So yes, this is correct. It’s also worth noting that this is our gradient because it’s our change in 𝑦 over our change in 𝑥 because for every hour that we go along, we add 60 dollars on. So, let’s demonstrate this. We could’ve chosen the two points on the line that were at one, 100 and two, 160. And if we look at the change in 𝑦 between these, it’s 60. And the change in 𝑥 is just one. So therefore, we’d have 60 over one, which would give us 60. So yes, it’s the same as our hourly labor charge. So that is our gradient.
Now for the final part of the question, we need to let 𝑦 be the cost in dollars for a job that takes 𝑥 hours. And we need to write an equation for 𝑦 in terms of 𝑥. So, what we’re gonna use is the general equation for a straight line. And that is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the gradient and 𝑏 is our 𝑦-intercept. Well, we already worked out the gradient and 𝑦-intercept. So therefore, we can say that our equation for 𝑦 in terms of 𝑥 is 𝑦 equals 60𝑥 plus 40.