2015 AP Calculus AB/BC 1d | AP Calculus AB solved exams | AP Calculus AB | Khan Academy

Part D. The pipe can hold 50 cubic feet of water before overflowing. For T greater than 8, water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Right, but do not solve an equation involving one or more integrals that gives the time W when the pipe will begin to overflow.

All right, so the pipe is going to overflow. We want to figure out the equation that gives the time W when the pipe begins to overflow. The pipe will begin to overflow when it crosses 50 cubic feet of water, you could say, right? When it hits 50 cubic feet of water, then it will begin to overflow.

So, we can figure out at what time the pipe has 50 cubic feet of water in it. We could just say, well, W of this time. I used uppercase W as my function for how much water is in the pipe, so capital W of lowercase w is going to be equal to 50, and so you would just solve for the W. And they say, right, but do not solve an equation.

Well, just to make this a little bit clearer, if D uppercase W of lowercase w is going to be 30 plus the integral from 0 to W. Actually, now, since I don't have T as one of my bounds, I could just say R of T minus D of T DT. So, let me just do that: R of T minus D of T DT.

So, this is the amount of total water in the pipe at time W. Well, this is going to be equal to 250. So, we have just written an equation involving one or more integrals that gives the time W when the pipe will begin to overflow.

So, if you could solve for W, that's the time that the pipe begins to overflow, and we are assuming that it doesn't just get to 50 and then somehow come back down. That it'll cross 50 at this time right here. You could test that a little bit more if you want. You could try slightly larger W or you could see that the rate that you have more flowing in than flowing out at that time, which is so this R of w is going to be greater than D of w.

So, it means you're only going to be increasing, so you're going to cross over right at that time. If you wanted that W, though, you'd solve this. Now, another option you could say, okay, we know W is going to be greater than 8. So you could say, okay, how much water do we have right at time equals 8? We figured that out in the last problem.

So you could say, at time equals 8, we have that much, 48.54. This is an approximation, but it's pretty close. Plus the amount of water we accumulate between time 8 and time W of R of T minus D of T DT is equal to 50. Either one of these would get you to the same place, the W at right when do we hit 50 cubic feet of water.

Then, if you wanted to test it further, you can make sure that your rate is increasing right at that or you have a net positive inflow of water at that point, which tells you that you're just about to start overflowing.