Find the limit as t approaches infinity of T(t).

I have this problem and it seems pretty easy. I think I'm just overthinking it.

At time t = 0 minutes, the temperature of a cup of coffee is 180 degrees Fahrenheit. Left in a room whose temperature is 70 degrees Fahrenheit, the coffee cools so that its temperature function T(t), also measure in degrees Fahrenheit, satisfies the differential equation: dT/dt = -1/2(T) + 35.

Find $\displaystyle \lim_{t \to \infty} T(t) $. Explain what this means in the context of the problem.

What is it asking me to do when it says "$\displaystyle \lim_{t \to \infty} T(t) $?" Is it asking me to set up an equation and solve (how would I do that?), or just explain what's happening? Also, can someone briefly explain what is happening? I'm stumped. Thanks for any help.

the limit as t approaches infinity of T(t)

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Originally Posted by maddawg579

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What is it asking me to do when it says "Find the limit as t approaches infinity of T(t)?" Is it asking me to set up an equation and solve (how would I do that?), or just explain what's happening? Also, can someone briefly explain what is happening? I'm stumped. Thanks for any help.

the limit as t approaches infinity of T(t)

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It's asking you to solve the differential equation, obtain the function $\displaystyle T(t)$, and then take it's limit as $\displaystyle t\to\infty$ right?

You can solve $\displaystyle \frac{dT}{dt}=-1/2 T+35$ by separating the variables: $\displaystyle \frac{dT}{35-1/2 T}=dt$

See if you can get to $\displaystyle \ln(35-1/2T)=-t/2+c$

then plug in the initial values $\displaystyle T(0)=180$ to determine what $\displaystyle c$ is, then take exp of both sides, (or take exp of both side then figure out what c is) to get it to the form $\displaystyle T(t)$, then take the limit $\displaystyle \lim_{t\to\infty} T(t)$.

So if I plug in 180 for T, I'd get $\displaystyle \ln(35-90)=-180/2+c$?

Then solve for c, which becomes 94.828, I think. Unless I skipped a step.

Then do I take the e of both sides? Is that what you mean by exp?

Dawg, you're getting the function name $\displaystyle T$ confused with the variable name $\displaystyle t$. Keep them straight. So if I have $\displaystyle \ln(35-1/2 T)=-t/2+c$ and $\displaystyle c$ is an arbitrary constant from the integration, then if I add, subtract, multiply anything to it, it's still a constant. Then doing a little algegra, I get:

$\displaystyle 35-1/2T=ce^{-t/2}$

(still just an arbitrary constant c even though I took exponentials.

little more:

$\displaystyle T(t)=70-ce^{-t/2}$

Now substitute the initial conditions $\displaystyle T(0)=180$ to figure out that c=-110 to get:

$\displaystyle T(t)=70+110 e^{-t/2}$

Now, you take that limit of that as $\displaystyle t\to\infty$.

Ok, that makes sense. I'm new to calc, so bear with me.

I can see you took the e of both sides, so the ln cancels with the e on left side, and on the right side the -t/2 becomes the exponent of e. But why is c multiplied to that?

After that I can see you moved 35 over and divided by -1/2 to get 70. I get that and I understand the rest.

$\displaystyle \ln(35-1/2 T)=-t/2+c$

so taking exponents of both sides:

$\displaystyle 35-1/2 T=e^{-t/2+c}=e^c e^{-t/2}$

but $\displaystyle e^c$ is still just some arbitrary constant if c is so I can replace it with just the arbitrary constant c and get:

$\displaystyle 35-1/2 T=ce^{-t/2}$

Ok, thanks so much. That all makes perfect sense. You were a huge help.

Wasn't it obvious from the start that $\displaystyle \lim_{t\to\infty} T(t)= 70$ degrees?

Mathematically, that "$\displaystyle -\frac{1}{2}$ multiplying the T in the differential equation tells you that the solution will involve an exponential with negative exponent that will go to 0 as t goes to infinity.

Physically, the coffee cools down to room temperature.

Oh I knew it had to equal 70 degrees in the end. I just needed to know the steps to get there.