2017 AP Calculus AB/BC 4c | AP Calculus AB solved exams | AP Calculus AB | Khan Academy

Let's now tackle part C, which tells us that for T is less than 10, an alternate model for the internal temperature of the potato at time T minutes is the function G that satisfies the differential equation:

The derivative of G with respect to T is equal to the negative of G minus 27 to the 2/3 power, where G of T is measured in degrees Celsius, and G of 0 is equal to 91.

Find an expression for G of T based on this model. What is the internal temperature of the potato at time T is equal to 3?

So, they gave us a differential equation; they want us to find an expression for G of T. So essentially, we need to find a solution to this differential equation, and then they want us to use that solution to find the internal temperature at time T equal to 3.

The first thing to appreciate is that this is in an AP Calculus class, and so if they're asking us to solve a differential equation or find a solution to a differential equation, it is unlikely to be a really strange differential equation. It's likely to be a separable differential equation. Once we find that solution, we just have to evaluate it at time T is equal to 3.

So let's rewrite the differential equation and then let's try to evaluate it. Let's see if we can find a solution.

The derivative of G with respect to T is equal to the negative of G minus 27 to the 2/3 power.

So, if this is going to be a separable differential equation, I want to separate the DG and the DT.

I'm going to treat my differentials like numbers or variables. I'm going to multiply both sides times the T differential, and so then I'm going to have the capital G differential DG is equal to - (G - 27) to the 2/3 power DT.

The whole notion here is I'm trying to get all of the G things on the side with the DG and then all the things that involve T on the side with the DT. So we don't see any T's here, so really we just have to get the G's over on this side while leaving the DT there. Then we can integrate both sides.

So, let's see if we were to divide both sides by G minus 27 to the 2/3. What do we have? Well, we can rewrite the left side as (G - 27) to the -2/3 DG is equal to -DT.

Actually, just to simplify, to make it a little bit more obvious, I could write that as -1 DT.

Now we just have to integrate both sides.

So we could rewrite this as let me write an integral sign. I'm going to integrate both sides here, and so what is this going to give me?

Well, on the left side here, you could try to do some U substitution saying U is equal to G minus 27, and then du would be DG. Or you might recognize that look, the derivative of (G - 27) is just DG; the derivative of (G - 27) is just going to be equal to one.

So you could even say, look, I have the derivative there, so I could really integrate with respect to G minus 27. Really, I would just use the reverse power rule. I would take my (G - 27). I would increment this exponent by 1, so see -2/3 + 1 is positive 1/3, so positive 1/3 power, and then I would divide by this new exponent.

Dividing by 1/3 is the same thing as multiplying by three, so that's the left side. This is going to be equal to the right-hand side, which is just going to be T plus C right over here.

So, how can we solve for C? Well, they give us some information; they say G of 0 is equal to 91.

When T is 0, G is 91.

So we can write, let me write it over here: 3 times (91 - 27) to the 1/3 power is equal to -0 + C.

Now, we're saying T is zero, so we could write negative 0 there, or we could just not write it and then plus C.

So, that's what C is going to be equal to. Let's see, 91 - 27 is 64; the 64 to the 1/3 power is 4.

So, we have C is equal to 12.

Now, let's see; we want to write an expression for G of T. So now let's just manipulate this and solve for G.

If we divide both sides by 3, we are going to get (G - 27) to the 1/3 power is equal to T/3 + 4.

Now I can take the cube of both sides, and I would get G minus 27 is equal to (T/3 + 4) to the 3rd power.

Now I just have to add 27 to both sides, and I'll make it clear: G as a function of T is equal to (T/3 + 4) to the 3rd power plus 27.

So I did the first part. This is an expression for G of T.

It was indeed a separable differential equation; it took a little bit of algebraic manipulation to get us there, but we were able to do it, and we were not only able to solve for the general solution, but we were able to find the particular solution using this initial condition right over here that G of 0 is equal to 91.

Now we'll do the easy part: based on this model, what is the internal temperature of the potato at time T equal to 3?

So G of 3 is equal to -(3/3) + 4 to the 3rd power plus 27.

Well, this is -1; so -1 + 4 is 3.

This is 3 to the 3rd power, which is 27.

Adding 27 gives us 54°C, and we are done.