Interpreting exponential functions

0:00- [Instructor] Okay, we're told function P gives the temperature of a slice of pizza in degrees Fahrenheit, m minutes after it comes out of the oven. The equation P of m is equal to 200 times 0.97 to the mth power defines this function. Here is the graph of P of m.

0:15So we can see it right over here, m minutes after you come out of the oven, you can figure out where the temperature is. So it looks like it starts at 200 degrees. And then once you get to about 20 minutes out, we're a little bit below, looks like 110 degrees.

0:30So the first question they ask us is, what is P of five? So pause this video and have a go at this. So you might be tempted to just look at five here, when after five minutes, and then you look up here and it looks like it's a little bit over 170. But just eyeballing it isn't gonna get you exactly where you need to get here.

0:51So to evaluate P of five, I actually want to go to the original function definition there, and I'm gonna need a calculator for that one. So I'm essentially going to take 0.97. So 0.97. I'm gonna take that to the mth power. In this case, m is five. So to the fifth power.

1:10And that's going to be equal to that. And then I need to multiply that times 200 to get P of five. So times 200 is equal to 171.74. So if I, let's just assume that that we can round this to the nearest whole unit. So this is roughly 172 degrees Fahrenheit. So I could write that here.

1:31172. And then they ask us a couple of more questions. In this situation, P of five represents, and before I even look at the interpretations, I just like to think, well, that's going to be the temperature in degrees Fahrenheit after the pizza has been five minutes out of the oven.

1:47Now let's see what the choices say. The temperature of the pizza in degrees Fahrenheit five minutes after it comes out of the oven. That one looks pretty good. Not the time it takes to cool completely. I guess that means get to room temperature. The original oven temperature, now that was probably 200 degrees.

2:02How fast the pizza cools per, none of those. That's what we just said. And I encourage you, before you look at the choices, just think, what would you say? The domain of P is, let's see, possible domains. All real numbers. So let's see, can you have a negative number in there?

2:19What does that make? What does it, we don't have any information on, we can't put a negative number into the function and get a reasonable answer. I think all, all negative, all non-negative numbers looks interesting. Let me look at the other choices. All numbers less than or equal to 200.

2:37No, we, could in theory, think about, well, what is the temperature of this pizza after 201 minutes? We could put that into that original equation that I just used. And it's not just whole numbers, we could put after five and a half minutes, we could put that in there into that equation we just used.

2:54So it's all non-negative numbers. That's the domain. And it represents, what does the domain represent? Well, it's time after taking it out of the oven. Let's see, the number of minutes since the pizza came out of the oven. Yeah, that's not the pizza's temperature at each time.

3:11We're just thinking about the domain, the number of minutes since the pizza came out of the oven. That's essentially this m right over here. It's not a rate, it's not the time that it was baked. We don't even know how long it was baked. So it's the time that the pizza came out of the oven.

3:24And we're done.