Linear equation word problems

- [Instructor] When Quinn returned from vacation, he turned the heat back on in his home. He set the temperature as high as it could go. Q represents the temperature in Quinn's home in degrees Celsius after t minutes. This Q is equal to 15 plus 0.4t. What was the temperature when Quinn returned from vacation? So pause this video and see if you can work this out on your own. All right, so they wanna know the temperature and you might get a little confused, hey maybe t is for temperature. No, t is time in minutes. Temperature is Q. Q represents the temperature. So they really wanna know is, what was Q when Quinn returned from vacation? Well, right when Quinn returned from vacation, that is when t is equal to zero. So this is equivalent to saying, what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to this original equation, we see that Q is equal to 15 plus 0.4 times the amount of elapsed time in minutes. So that's times zero. So that's just going to be 15 degrees Celsius. If you're familiar with slope intercept form, you could think of it as our temperature is equal to 0.4 times the elapsed time plus 15. So t equals zero, you're left with just this term which in many cases we view as our y-intercept. What is going on right when we're just getting started? Right when our horizontal variable is equal to zero. And our horizontal variable in this situation is elapsed time. How much do the temperature increase every minute? There's a couple of ways you can think about this. One if you recognize, this is slope intercept form. You could see that 0.4 is the slope. So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. So you could do it that way. You could try out some values. You could say, all right, let me think about what Q is going to be based on t. So time, t equals zero right when you got home. We're to figure out that the temperature is 15 degrees Celsius. At t equals one, what happens? It's going to be 15 plus 0.4 times one. That's just gonna be 15.4. Notice, when we increased our time by one, our temperature increased by 0.4 degrees Celsius by the slope. And what happened again if we increased time by another minute, if we go from one to two, we would get two, 15.8. We would increase temperature by another 0.4. How much will the temperature increase if Quinn leaves the heat on for 20 minutes? Pause the video and see if you can solve that. All right, now we have to be careful here. They're not asking us what is the temperature after 20 minutes. They're saying, how much will the temperature increase if he leaves the heat on for 20 minutes? If we just wanna know what is the temperature after 20 minutes, we would just say, okay, what is Q when t is equal to 20? So it'd be 15 plus 0.4 times 20. 0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsius after 20 minutes. But that's not what they're asking us. They're asking, how much will the temperature increase? Well, where did we start from? We started from 15 degrees Celsius and now after 20 minutes we have gone to 23 degrees Celsius. So we have increased by eight degrees Celsius. Or, another way to think about it, we have increased by this amount right over here. We started at 15 and after 20 minutes, we have increased by 0.4 times 20 which is eight degrees Celsius. We're done!