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# differential equation Watch

1. a cup of tea cools at a rate proportional to the temp of the tea above that of the surrounding air. initially, the tea is at 95 degrees celcius and cools at a rate of 0.5 cs^-1. the surrounding air is at 15 degrees celcius. find a differential equation to model this situation?

I got - dtheta/dt is proportional to (theta-15)

which turns into dtheta/dt = - (theta-15)

but the answer says it is dtheta/dt = -[(theta-15]/160]

I know the 160 is because (95-15)/0.5

but why?
2. (Original post by 0utdoorz)
a cup of tea cools at a rate proportional to the temp of the tea above that of the surrounding air. initially, the tea is at 95 degrees celcius and cools at a rate of 0.5 cs^-1. the surrounding air is at 15 degrees celcius. find a differential equation to model this situation?

I got - dtheta/dt is proportional to (theta-15)

which turns into dtheta/dt = - (theta-15)

but the answer says it is dtheta/dt = -[(theta-15]/160]

where do I get the 160 from?
You are missing a constant in the equality.
The change in temperature is the 0.5 celsius/s
3. (Original post by 0utdoorz)
I got - dtheta/dt is proportional to (theta-15)

which turns into dtheta/dt = - (theta-15)
Can you post working to show how you got from the first line to the second line?
4. (Original post by 0utdoorz)
a cup of tea cools at a rate proportional to the temp of the tea above that of the surrounding air. initially, the tea is at 95 degrees celcius and cools at a rate of 0.5 cs^-1. the surrounding air is at 15 degrees celcius. find a differential equation to model this situation?

I got - dtheta/dt is proportional to (theta-15)

which turns into dtheta/dt = - (theta-15)

but the answer says it is dtheta/dt = -[(theta-15]/160]

where do I get the 160 from?
Well you're missing the constant of proportionality. But I got 1/160, not 160. So I've done something wrong, maybe you can spot the mistake, I don't know what it is. But this is what I got, and it's something along these lines.

Im just calling theta x here to make it easy,
but if
dx/dt ∝ (x - 15)
dx/dt = -k(x-15)
As tea is initially at 95 degrees and rate is 0.5
0.5 = (-k)(95-15)
0.5 = (-k)(80)
0.5/80 = -k

Then plug into the equation above.
5. (Original post by BooAlphie)
Well you're missing the constant of proportionality. But I got 1/160, not 160. So I've done something wrong, maybe you can spot the mistake, I don't know what it is. Did the question say proportional or inversly proportional, coz If it's inversely proportional, what I've done is right, just 1/it.
But you've forgotten k.

Im just calling theta x here to make it easy,
but if
dx/dt ∝ (x - 15)
dx/dt = -k(x-15)
As tea is initially at 95 degrees and rate is 0.5
0.5 = (-k)(95-15)
0.5 = (-k)(80)
0.5/80 = -k

Then plug into the equation above.
You're correct and this is the same as the OP's given answer. Have another look.
6. thanks! I understand what I've done wrong now
7. (Original post by notnek)
You're correct and this is the same as the OP's given answer. Have another look.
I just did a past paper and had a question like this and realised I was right and came back to correct it, you beat me to it

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