Constructing rational function models
About this video
A family drives a fixed distance d, and the trip time T is inversely proportional to the average speed s. Using distance = rate x time, the video writes T(s) = d/s, then uses the data point s = 30, T = 12 to find d = 360, giving T(s) = 360/s.
You'll be able to
- explain what it means for one quantity to be inversely proportional to another.
- build a rational model T(s) = d/s from distance = rate x time.
- use a data point to find the constant in an inverse proportion model.
Key words
- inversely proportional
- When one quantity goes up, the other goes down so that their product stays the same, as in T = d/s.
- average speed
- The total distance divided by the total time taken.
- rational function
- A function written as a fraction with a variable in the denominator, such as 360/s.
Transcript
- [Instructor] We're told a family is driving to a vacation destination that is a fixed distance away, d. The time the trip takes in hours, capital T, is inversely proportional to the average speed in miles per hour, s, of the car. Yeah, that makes sense. The faster you go, the less time it's going to take, or the more time it took, the slower you went.
So inverse proportionality makes sense. Which equation models the time taken as a function of the average speed? Pause the video, have a go at that. Okay, so let's just remind ourselves, we could say that distance is equal to rate times time. In this example, rate is given by the letter s, time is capital T.
If we wanted to express time as a function of s, we could divide both sides by s, and we would get, let me put a little arrow here, time is equal to our distance divided by s. Or we could write T of s is equal to distance divided by s. Actually, let me just write that.
I could write T of s, it's a function of s, is equal to distance divided by s. So what we could do is we could actually use two of these points to solve for what this distance is going to be. For example, when s is 30, our time is 12. So we could just substitute that in right over here.
So our time is 12 when our speed is 30, so d divided by 30. And so if I were to multiply both sides by 30 to solve for d, multiply both sides by 30, I get that 360 is equal to d. And so we have that T of s is equal to 360 divided by s, which is this choice right over here.
Now you could have immediately removed some of these choices because in choice A, T isn't inversely proportional with s, it's proportional. So we don't like that. That would apply the faster you go, the more time it takes. Doesn't make sense. Similarly, you could rule this one out as well.
And then you could have actually done it the other way. You could have looked at these two choices and seen which one matches the data, but it wasn't too bad to just say, all right, let's substitute for T and s and solve for d.
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