Linear equation word problems
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. They say 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 want to know the temperature, and you might get a little confused. Say, "Hey, maybe T is for temperature?" No, T is time in minutes; temperature is Q. Q represents the temperature, so what they really want to 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 does the temperature increase every minute?
There's a couple of ways you could think about this. One, if you recognize this as 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, or 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 he got home. We already figured out that the temperature is 15 degrees Celsius.
At T equals one, what happens? Well, it's going to be 15 plus 0.4 times 1. Well, that's just going to be 15.4. Notice when we increased our time by 1, our temperature increased by 0.4 degrees Celsius, by the slope.
And it would happen again if we increased time by another minute. If we go from 1 to 2, we would get to 15.8. We would increase the 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 answer 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 want to 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 8. 8 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, what 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 8 degrees Celsius, or another way to think about it is we have increased by this amount right over here. We started at 15, and after 20 minutes, we've increased by 0.4 times 20, which is 8 degrees Celsius. And we're done.