Differentiate equation!!! Really need help!

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thegallowsgirl
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#1
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HELP!
I have this question, and I've got about 20 different answers to it but I'm completely lost.

T= a+b e^-kt

T is the temperature at time t, and a, b and k are constants with values a = 20 °C, b = 80 °C and k = 2.6 × 10^-4 s^-1.

Differentiate the equation with respect to t and thus give values (with appropriate units and to two significant figures) for the rate of change of temperature with time at times of 1.0 × 103 s and 5.0 × 103 s.

Can anybody please help me with the steps in doing this please??
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Zacken
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(Original post by thegallowsgirl)
HELP!
I have this question, and I've got about 20 different answers to it but I'm completely lost.

T= a+b e^-kt

T is the temperature at time t, and a, b and k are constants with values a = 20 °C, b = 80 °C and k = 2.6 × 10^-4 s^-1.

Differentiate the equation with respect to t and thus give values (with appropriate units and to two significant figures) for the rate of change of temperature with time at times of 1.0 × 103 s and 5.0 × 103 s.

Can anybody please help me with the steps in doing this please??
Constants, go away, obviously.

\frac{d}{dt}(e^{-kt}) = -k e^{-kt}.
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thegallowsgirl
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So far I have

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4)
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thegallowsgirl
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(Original post by Zacken)
Constants, go away, obviously.

\frac{d}{dt}(e^{-kt}) = -k e^{-kt}.
okay, so if I am understanding correctly (which I may not be sorry) then:
-k e^-kt would be:
(-2.6 x 10^-4) x (-2.6 x 10^4) x (t = 1 x 10^3) ?
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DPD99
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Internet search came up with http://mymathforum.com/algebra/33032...n-problem.html

It may be some help.
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Zacken
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(Original post by thegallowsgirl)
So far I have

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4)
Where'd the small t in the power go?
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PossiblyNotGod
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Is this T= (a+b) x (e^-kt)?
or T= a+b + (e^-kt)?
or T= a+(b*e^-kt)?
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thegallowsgirl
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(Original post by Zacken)
Where'd the small t in the power go?
Sorry, that was a typing error.

dT/dt = (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4) x t

= (20 + 80) x (-2.6 x 10^-4)e ^(-2.6 x 10^-4) x (1 x 10^3)

With constants a & b removed:

= (-2.6 x 10^-4)e ^(2.6 x 10^-4) x (1 x 10^3)

Quick question - e is normally 2.718, but I'm unsure as to whether I need to include that in this equation?
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thegallowsgirl
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(Original post by XOR_)
Is this T= (a+b) x (e^-kt)?
or T= (a+b) + (e^-kt)?
or T= a+(b*e^-kt)?
It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.
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Zacken
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(Original post by thegallowsgirl)
It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.
I would presume it's a + [be^(-kt)].
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1420787
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(Original post by thegallowsgirl)
It's written T = a+b e^-kt so I presume it's T= (a+b) x (e^-kt)? At least that's what I've been doing.
I presume it would be as written, without the brackets.

The questions looks set up to calculate temperature change of someting in a room of temperature 20 degrees. As time goes on the temperature will tend to 20 degrees, as is natural.
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thegallowsgirl
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(Original post by Zacken)
I would presume it's a + [be^(-kt)].
That could be where I'm going wrong.
If I did it that way, it would be:

20 + [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000) ?
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Zacken
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(Original post by thegallowsgirl)
That could be where I'm going wrong.
If I did it that way, it would be:

20 + [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000) ?
No, when you differentiate it, the a disappears.
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thegallowsgirl
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(Original post by Zacken)
No, when you differentiate it, the a disappears.
Right!
So it would be [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000)] without the 20 in front.
So calculating it would result in 0.005408 = 5.4 x 10^-3
Is that in the right area or is there a step that I've missed out? Sorry for asking so many questions, I'm trying to understand the steps and how they result in the answer.
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Zacken
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(Original post by thegallowsgirl)
Right!
So it would be [80 x (-2.6 x 10^-4)e x (-2.6 x 10^-4)x (1000)] without the 20 in front.
So calculating it would result in 0.005408 = 5.4 x 10^-3
Is that in the right area or is there a step that I've missed out? Sorry for asking so many questions, I'm trying to understand the steps and how they result in the answer.
It would be (stuff in front seems fine) but the e^{-2.6\times 10^{-4} \times 1000}.

That is, e raised to the power of the the number in the exponent, not e * number.
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thegallowsgirl
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Thank you so much! I was thinking I'd missed something out.
[80 x (-2.6 x 10^-4)e^ (-2.6 x 10^-4)x (1000)] I'm trying to figure out how I can input it into a calculator now. Thank you for your help!!
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Zacken
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(Original post by thegallowsgirl)
Thank you so much! I was thinking I'd missed something out.
[80 x (-2.6 x 10^-4)e^ (-2.6 x 10^-4)x (1000)] I'm trying to figure out how I can input it into a calculator now. Thank you for your help!!
Type in ((-2.6 x 10^-4)x (1000)). Press equals.

Now type in e^(ANS). Press equals.

Now do Ans * 80 *(-2.6 * ...)
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