00:01
All right, in your question you're given the function h of t, which is representing of newton's law of cooling, and it has to do with a cup of coffee being handed to a person at a park, and it's describing the temperature after a certain amount of time.
00:18
Now, i looked through this question, and it never states what time is measured in, but what logically makes sense is for time to be measured in minutes for this problem.
00:30
So we're going to go with minutes anytime we describe time, because i don't see any reference to minutes or seconds or hours.
00:39
All right, part a, you're asked to determine is this a beverage, a hot coffee or an iced coffee? well, we can figure out what the temperature is at time zero by plugging a zero in for t, and that means h of zero is going to be 93 times 0 .91 to the zero power plus 68.
01:03
Now, anything raised to the zero power other than zero is one, so this becomes 93 plus 68, or 161 degrees.
01:15
So i would say that this is a hot coffee.
01:23
Your explanation could be 161 degrees is definitely not iced.
01:39
All right, moving on to part b, you're asked what is the asthma toad of the graph, and what does it mean in the context of this question? the asthma toad would be what temperature we approach after a long period of time, so an infinite amount of time.
01:54
Imagine plugging infinity in here.
01:56
As t gets larger and larger, we're taking this 0 .91 raised to a higher power.
02:02
It's actually going to decrease the size of the number, moving it closer to zero, which is going to eventually cause the 93 not to be part of the problem.
02:12
We can identify in these questions the asthma toad then as this number.
02:18
So we have a horizontal asthma toad at y equals 68, and what that would mean, it's the temperature that we're cooling to.
02:29
We would never actually reach it, but we're saying that is your outside air temperature.
02:40
Basically, that coffee can't get colder than it is outside, which is the 68 outside air temperature.
02:47
Now in part c, you're to sketch a rough graph of this function, so we know at time 0, it's 161 degrees.
02:59
So i'm going to go ahead and i'm going to put a dot here for 161.
03:04
We're only to include times greater than 0, so i'm showing quadrant 1.
03:09
I'm also going to put a line here at 68, or a spot at 68.
03:16
What we would see is our temperature dropping rapidly, but then slowing down and approaching that horizontal asthma toad at 68.
03:26
And i would say that works as a rough graph.
03:30
Part d, we're to calculate the coordinates of the y -intercept.
03:35
We already know those from part a.
03:38
The coordinates of the y -intercept would be 0 for t and 161 for the y -coordinate.
03:48
And what does that mean in the context of the question? i would say the coffee was 161 degrees when handed to you, because the question is stating an employee hands it to you.
04:19
All right, part e, we want to know what the range of h of t.
04:25
In reality, we're not going to go in time that's negative, where the graph would have possibly went higher.
04:31
So we're only looking at time t greater than or equal to 0.
04:36
The range then would be the lowest amount we could approach, and i would say we would never get there.
04:45
So i would use a parenthesis on 68, and then our highest possible temperature is 161 that we did start at, so it gets a bracket.
05:00
In part f, you're asked to find h of 10.
05:04
Oops, sorry.
05:06
Part e said, what does meaning does the range have? this is the possible temperatures for the coffee over time.
05:30
All right, now on to part f, we want to find h of 10.
05:34
And to find h of 10, we're just going to go back to our original equation, and we're going to plug a 10 in for the t, and we can type that into our calculator...