Thread: 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 . Explain what this means in the context of the problem.

What is it asking me to do when it says " ?" 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)

Originally Posted by maddawg579

I
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)

It's asking you to solve the differential equation, obtain the function , and then take it's limit as right?

You can solve by separating the variables:

See if you can get to

then plug in the initial values to determine what 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 , then take the limit .

So if I plug in 180 for T, I'd get ?

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 confused with the variable name . Keep them straight. So if I have and 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:



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

little more:



Now substitute the initial conditions to figure out that c=-110 to get:



Now, you take that limit of that as .

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.

so taking exponents of both sides:



but is still just some arbitrary constant if c is so I can replace it with just the arbitrary constant c and get:

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

Wasn't it obvious from the start that degrees?

Mathematically, that " 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.