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Modeling with multiple variables: Ice cream

Modeling the relationship between three quantities (or more) isn't that different from modeling the relationship between two quantities. Here is an example of a model that relates different quantities related to two people walking to the same ice cream shop. Created by Sal Khan.

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Video transcript

- [Instructor] We're told that Ben's home is x kilometers from an ice cream shop. Jerry's home is y kilometers from the same shop. Then it tells us they each left their home at the same time and met at the ice cream shop at the same time. Ben walked an average speed, let me do this in a new color, average speed of five kilometers per hour. And Jerry rode his bicycle at an average speed of v kilometers per hour. Write an equation that relates x, y, and v. Pause this video and see if you can do that. All right, so let's just remind ourselves how distance, speed, and time are related. You might be familiar with things like distance is equal to rate times time, or another way you could think about it is you could write the distance is equal to speed times time. Or if you wanna solve for time, you can divide both sides by speed. So you could get distance over speed, over speed, is equal to time. Now the reason why I set it up this way is that we know that Ben's time and Jerry's time is the same. They covered maybe different distances at maybe different speeds, but it took them the exact same amount of time. So Ben's distance divided by Ben's speed should be the same as Jerry's distance divided by Jerry's speed. So let me write that down. So Ben's distance, Ben's distance, divided by Ben's speed, and let me write it in this color, Ben's speed should be equal to Jerry's distance, Jerry's distance, divided by Jerry's speed, Jerry's speed. Now which of these do we know or do we already have variables to find? Well we know that Ben's distance from the ice cream shop is x, so this is represented by x. We know that Jerry's distance from the ice cream shop is represented by y, so this is y. we know that Ben's speed is five kilometers per hour. So we're assuming everything is in kilometers per hour. So this would be five. And then Jerry's speed is v kilometers per hour. So this is v right over there. And so we could rewrite all of this as x over five is equal to y over v. And once again, the way that I set this up, the left side is the amount of time Ben takes to get to the ice cream shop. This is, on the right hand side, this is the amount of time Jerry takes to get to the ice cream shop. And they tell us it's the same amount of time. So there you have it. We have an equation that relates x, y, and v. And they gave us the five. Now it's completely possible that instead of the five, they gave us something else and Ben's speed was the variable. If they did that, then we would have a different given and maybe a different variable, but the structure of our equation would be the same, that Ben's distance divided by Ben speed's would need to be equal to Jerry's distance divided by Jerry's speed.