Topic B: Unit rate and constant of proportionality
- [Voiceover] Let's get some practice interpreting graphs of proportional relationships. This says the proportion relationship between the distance driven and the amount of time driving shown in the following graph, so we have the distance driven on the vertical axis, it's measured in kilometers, then we have the time driving and it's measured in hours along the horizontal axis. We can tell just visually that this, indeed, is a proportional relationship, how do we know that? Well, the point (0, 0) is on this graph, the graph goes through the origin. If we have zero time, then we have zero distance, and we can also see that it's a line, then it's a linear relationship. If you have a linear relationship that goes through the origin, you're dealing with a proportional relationship. You could also see that by taking out some points here. Let's see, I'm just eyeballing, so I'm gonna look at where does the graph kind of hit a very well-defined point. Actually, point A right over here, we see that when our time is five hours, our distance travelled or driven is 400 km. Then if we look at time, this point right over here, when our time is 2.5, we see that our distance driven is 200 km. And notice, the ratio between these variables at any one of these points is the same. 400 divided by five is 80, and 200 divided by 2.5 is also going to be 80. Or if you wanna go the other way around, to go from time to distance, we're always multiplying by 80. In fact, we can say that distance divided by time, our proportionality constant is going to be 80. Or if we wanted to include the units there, it might be a little more obvious than dealing with the rate, distance is in kilometers, time is in hours, 80 km per hour, this is the rate at which we are driving. We are going at this speed, 80 km per hour. This is also the proportionality constant. Anyway, with all of that out of the way, let's actually answer the questions. Which statements about the graph are true? Select all that apply. The vertical coordinate of point A represents the distance driven in four hours. The vertical coordinate. So point A is at the coordinate (5, 400). The vertical coordinate tells us how high to go up, how far to move in the vertical direction. That's gonna be the second coordinate right over here, so this is the vertical coordinate. This right over here tells us the distance we've driven in four hours, so yes, the vertical coordinate of point A represents the distance driven in four hours. We've driven 400 km, I like that one, I'll check that one. The distance driven in one hour is 80 km. When you just try to eyeball it off of the graph here, I see, after one hour, it looks like it's a little bit at more than 75, so yeah, 80 seems like a reasonable one. But we see it even more clearly when we look at the calculation that we did. If you take a distance, any one of these distance, 400 km over five hours, tells you that your rate is 80 km per hour, and you know this is going to happen for any point on this line, they tell us it's a proportional relationship, and we can tell, visually, it's a proportional relationship. It goes through the origin and it is a line. So the distance travelled in one hour is going to be 80 km 'cause we're going 80 km per hour. So in one hour, we're gonna go 80 km. So I like that choice as well, of course, I won't pick none of the above because I found two choices that I liked. Let's do another one of these, this is fun. A grocery story sells cashews. The relationship between weight and cost of cashews is shown in the following graph, and once again, we see it is a proportional relationship. It goes through the origin, and it is a line. Which statements about the graph are true? Select all that apply. The point (0, 0) shows the cost. It's $0 for 0 kg of cashews. Yeah, 0 kg of cashews, it's $0. Yeah, that makes sense. The point (2, 60), that's this point right over here, shows the cost is $2 for 60 kg of cashews, the cost is $2 for 60 kg of cashews. No, this is showing us that the cost is $60 the cost is $60 for 2 kg of cashews. This axis right over here is weight, actually, if you're measuring kilograms, you're really talking about mass, but I don't wanna get too particular here. You have a mass of 2 kg, I guess you could say the horizontal coordinate tells you the mass. The vertical coordinate tells you the cost, and that is $60. So it's $60 for 2 kg, not $2 for 60 kg. That would be a deal, if you could get, this is what a medium-sized grown man would weigh 60 kg, and $2 for that much cashews, that would be an incredible deal. So you definitely wanna rule that one out. And I'm not gonna pick none of the above because I already figured that I liked the first choice up there. Let's just do one more, just for good measure. An employee earns an hourly wage shown in the graph below. Find the hourly wage. We see here in the horizontal axis, we have hours worked, and in the vertical axis, earnings. We see it's a proportional relationship. And that makes sense, the ratio between earnings and hours should always be constant, if we're earning money at a constant rate, we see it visually, we go through the origin and we're dealing with a line. So there's a bunch of ways that we could look at it. We could say hours, hours, then dollars. After one hour, we've made $40. And just the fact, visually, that we know it's a proportional relationship says we're making $40 per hour. If we take dollars divided by hours, I'll write out the units, 40 dollars divided by one hour. This right over here is 40 dollars per hour. You could view this as our proportionality constant, or you could use the rate in which we're earning money, $40 per hour, and you see that. Two hours, $80. 80 divided by two is 40. Three hours, $120. 120 divided by three is 40. Our proportionality constant is 40. That your earnings divided by hours is always going to be equal to 40, or you could say that your earnings is equal to, if you multiply both sides by hours, is going to be equal to 40 times your hours. Your earnings in dollars is going to be equal to 40 times your hours.