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2015 AP Calculus AB/BC 1ab

Water in a pipe.

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  • blobby green style avatar for user Mark Yahraes Johnson
    at , you calculated the answer in radians. Why did you use radians and how do you know when to use radians or degrees?
    (5 votes)
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    • old spice man green style avatar for user Jonathan Ziesmer
      When in doubt, assume radians. Almost all mathematicians use radians by default.

      You can tell the difference between radians and degrees by looking for the ° symbol. If the numbers of an angle measure are followed by a °, it will be degrees. Otherwise it will always be radians.

      I hope this helps!
      (5 votes)
  • piceratops ultimate style avatar for user abchess2000
    In part one, wouldn't you need to account for the water blockage not letting water flow into the top because its already full?
    (2 votes)
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    • aqualine seed style avatar for user ishikamukerji
      The blockage is already accounted for as it affects the rate at which it flows out. It does not specifically say that the top is blocked, it just says its blocked somewhere. That blockage just affects the rate the water comes out. That is why there are 2 different equations, I'm assuming the blockage is somewhere inside the pipe.
      (2 votes)
  • blobby green style avatar for user altamiranod21
    What about the initial 30 cubic feet of water when t = 0 ?
    (2 votes)
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  • primosaur seedling style avatar for user Kelsey O'Haire
    In part A, why didn't you add the initial variable of 30 to your final answer?
    (1 vote)
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  • blobby green style avatar for user Yashas Acharya
    For part b, since the d(t) and r(t) indicates the rate of flow, why can't we just calc r(3) - d(3) to see the whether the answer is positive or negative?
    (1 vote)
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  • leaf green style avatar for user rben2004
    Would it be possible to find the definite integral in part(a) by hand? If so what would it be?
    (1 vote)
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  • duskpin ultimate style avatar for user Akash Arunabharathi
    Is there a way to merge these two different functions into one single function?
    (1 vote)
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  • purple pi teal style avatar for user turaevshokhrukh7
    Can someone help me out with this question: Suppose that a function f(x) satisfies the relation (x^2+1)f(x) + f(x)^3 = 3 for every real number x. Evaluate f'(1) .
    i would really be grateful if someone could post a solution to this question.
    i'm quite confused
    (1 vote)
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  • duskpin tree style avatar for user Graciela Maldonado
    How do you know when to put your calculator on radian mode?
    (1 vote)
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    • primosaur ultimate style avatar for user Julia Pockat
      Usually for AP calculus classes you can assume that your calculator needs to be in radian mode unless otherwise stated or if all of the angle measurements are in degrees. I don't think I can recall a time when I was asked to use degree mode in calc class, except for maybe with some problems involving finding lengths of sides using tangent, cosines and sine. I hope this helped!
      (1 vote)
  • blobby green style avatar for user Rafik Belhaouas
    The result of question a should be 76.57 cubic feet. Sorry for nitpicking but stating what is the unit is very important.
    (0 votes)
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Video transcript

- [Voiceover] The rate at which rainwater flows into a drainpipe is modeled by the function R, where R of t is equal to 20sin of t squared over 35 cubic feet per hour. t is measured in hours and 0 is less than or equal to t, which is less than or equal to 8, so t is gonna go between 0 and 8. The pipe is partially blocked, allowing water to drain out the other end of the pipe at rate modeled by D of t. It's equal to -0.04t to the third power plus 0.4t squared plus 0.96t cubic feet per hour. For the same interval right over here, there are 30 cubic feet of water in the pipe at time t equals 0. Alright, Part A. How many cubic feet of rainwater flow into the pipe during the 8 hour time interval 0 is less than or equal to t is less than or equal to 8? Alright, so we know the rate, the rate that things flow into the rainwater pipe. Let me draw a little rainwater pipe here just so that we can visualize what's going on. So if that is the pipe right over there, things are flowing in at a rate of R of t, and things are flowing out at a rate of D of t. And they even tell us that there is 30 cubic feet of water right in the beginning. But these are the rates of entry and the rates of exiting. So they're asking how many cubic feet of water flow into, so enter into the pipe, during the 8-hour time interval. So if you have your rate, this is the rate at which things are flowing into it, they give it in cubic feet per hour. If you multiply times some change in time, even an infinitesimally small change in time, so Dt, this is the amount that flows in over that very small change in time. And so what we wanna do is we wanna sum up these amounts over very small changes in time to go from time is equal to 0, all the way to time is equal to 8. So this expression right over here, this is going to give us how many cubic feet of water flow into the pipe. Once again, what am I doing? R of t times D of t, this is how much flows, what volume flows in over a very small interval, dt, and then we're gonna sum it up from t equals 0 to t equals 8. That's the power of the definite integral. And so this is going to be equal to the integral from 0 to 8 of 20sin of t squared over 35 dt. And lucky for us we can use calculators in this section of the AP exam, so let's bring out a graphing calculator where we can evaluate definite integrals. And so let's see. We wanna do definite integrals so I can click math right over here, move down. So this function, fn integral, this is a integral of a function, or a function integral right over here, so we press Enter. And the way that you do it is you first define the function, then you put a comma. Then you say what variable is the variable that you're integrating with respect to. And then you put the bounds of integration. So I'm gonna write 20sin of and just cuz it's easier for me to input x than t, I'm gonna use x, but if you just do this as sin of x squared over 35 dx you're gonna get the same value so you're going to get x squared divided by 35. Close that parentheses. So that is my function there. Actually, I don't know if it's going to understand. Let me put the times 2nd, insert, times just to make sure it understands that. Ok, so that's my function and then let me throw a comma here, make it clear that I'm integrating with respect to x. I could've put a t here and integrated it with respect to t, we would get the same value. Comma, my lower bound is 0. And my upper bound is 8. And then close the parentheses and let the calculator munch on it a little bit. And we get 76.570 so this is approximately Seventy-six point five, seven, zero. Now let's tackle the next part. Is the amount of water in the pipe increasing or decreasing at time t is equal to 3 hours? Give a reason for your answer. Well, what would make it increasing? Well if the rate at which things are going in is larger than the rate of things going out, then the amount of water would be increasing. But if it's the other way around, if we're draining faster at t equals 3, then things are flowing into the pipe, well then the amount of water would be decreasing. Let me be clear, so amount, if R of t greater than, actually let me write it this way, if R of 3, t equals 3 cuz t is given in hour. t is measured in hours. If R of 3 is greater than D of 3, then D of 3, If R of 3 is greater than D of 3 that means water's flowing in at a higher rate than leaving. So that means that water in pipe, let me right then, then water in pipe Increasing. Increasing. And then if it's the other way around, if D of 3 is greater than R of 3, then water in pipe decreasing, then you're draining faster than you're putting into it. Then water in pipe decreasing. decreasing. So we just have to evaluate these functions at 3. So let's see R. Actually I can do it right over here. So let me make a little line here. R of 3 is equal to, well let me get my calculator out. This is going to be, whoops, not that calculator, Let me get this calculator out. And I'm assuming that things are in radians here. So I already put my calculator in radian mode. So it's going to be 20 times sin of 3 squared is 9, divided by 35, and it gives us, this is equal to approximately 5.09. So this is approximately 5.09 and D of 3 is going to be approximately, let me get the calculator back out. So it is, We have -0.04 times 3 to the third power, so times 27, plus 0.4 times 9, times 9, t squared. plus 0.96 times t, times 3. And this gives us 5.4. So this is equal to 5.4. So D of 3 is greater than R of 3, so water decreasing. We're draining faster than we're getting water into it so water is decreasing.