# Thread: Exponential Cooling Curve Problem

1. ## Exponential Cooling Curve Problem

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2. Originally Posted by Jaydee
Hello,

I have some review to work on over the holidays for grade 11 math. This year math and I haven’t been getting along as well as we usually do, however, and so I’m having some trouble with our review for our upcoming exam, and I really need to have some questions explained out for me as to how I can do them so that I can study them over the break. I’ve done most of them, but here is one that I’m having trouble with:
Coffee, at 95 C, is poured into a cup. After 2 minutes the coffee cools to 90 C. The room temperature is kept at 20 C. Assuming an exponential cooling curve, how long will it take for the coffee to reach 80 C?

I figured that the equation would look something like: 95(?)^t/? + 20
However I can’t figure out what goes in brackets and what is the denominator for the t. This is how we typically do cooling curves in class, though we’ve never had to write our own equations.

Any help would be much appreciated,

JD
Your model is wrong. The model to use is $T = a (b)^t + 20$ (other models, such as $T = a e^{kt} + 20$, could also be used).

Substitute T = 95 at t = 0: 95 = a + 20 => a = 75.

Update the model (just like Ronnie): $T = 75 (b)^t + 20$.

Now substitute T = 90 when t = 2 and solve for b.

3. Ah, I see what I did wrong.

So ...

T = 75(b)^t + 20

90 = 75(b)^2 + 20
90 - 20 = 75(b)^2
70 / 75 = b^2
±√70/75 = b
b = ±0.97 ?

4. Originally Posted by Jaydee
Ah, I see what I did wrong.

So ...

T = 75(b)^t + 20

90 = 75(b)^2 + 20
90 - 20 = 75(b)^2
70 / 75 = b^2
±√70/75 = b
b = ±0.97 ?
Does a negative value of b make sense?

5. Originally Posted by mr fantastic
Does a negative value of b make sense?
Now that I'm not too sure of. Is 'b' the rate of cooling? If so, wouldn't it be negative?

We never really discussed exponential cooling curves in class, so my knowledge of them is pretty ... limited. I apologize in advance.

6. Originally Posted by Jaydee
Now that I'm not too sure of. Is 'b' the rate of cooling? If so, wouldn't it be negative?

We never really discussed exponential cooling curves in class, so my knowledge of them is pretty ... limited. I apologize in advance.
You don't need any special knowledge of exponential curves - just some mathematical common sense. Let's say you use the negative value of b.

Substitute some value of t into the rule. Do you get values of T that make sense? What happens if you substitute t = 1/2, for example ...? What values of T do you get for odd values of t ....?

Use some mathematical common sense.

7. Oh, I guess -0.97 wouldn't make sense. It seems as though the temperture fluctuates too wildly.
At 2 hours I came up with: T = 90.6
Then at 3 hours it went to: T = -48.5
Then at 4: T = 86.4
That doesn't make a heck of a lot of sense. Especially when a half hour yields a 'domain error' on my calculator.

So b has to be positive, then...

(And I apologize again. I'm pretty much devoid of mathematical common sense. It's never made any sense to me...)