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anonymous

  • 5 years ago

Oil is leaking from a tanker at the rate of R(t) = 2000e^(-0.2t) gallons per hour where t is measured in hours. How much oil leaks out of the tanker from the time t = 0 to t = 10

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  1. anonymous
    • 5 years ago
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    For background knowledge, what math are you in?

  2. anonymous
    • 5 years ago
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    AP Calculus

  3. anonymous
    • 5 years ago
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    Okay, that's what I thought. So you know integrals?

  4. anonymous
    • 5 years ago
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    Sort of

  5. anonymous
    • 5 years ago
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    Do you know what they're used for?

  6. anonymous
    • 5 years ago
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    not really

  7. anonymous
    • 5 years ago
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    Well, this is one prime example of what they're used for. If you graphed the equation you said, taking the integral of it would essentially find the area filled in between your line and the x axis

  8. anonymous
    • 5 years ago
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    What this means when the line represents a rate is that your x axis will essentially represent time elapsed and the area filled in is the total

  9. anonymous
    • 5 years ago
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    Do you get that?

  10. anonymous
    • 5 years ago
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    Yeah, I think so! Can I just plug the equation into the calculator to graph it and then get my answer from there?

  11. anonymous
    • 5 years ago
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    Not really (unless you know have a TI-89 which can integrate)

  12. anonymous
    • 5 years ago
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    I do have one :)

  13. anonymous
    • 5 years ago
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    Hahaha. Well if you're allowed to, go for it. Before I say do it to it, how would you set up the integral for this equation?

  14. anonymous
    • 5 years ago
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    Also, just thought you should know that sunflowers are my dad's favorite so I approve of the name, haha

  15. anonymous
    • 5 years ago
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    \[\int\limits_{0}^{10}2000e^(-.2t)\]

  16. anonymous
    • 5 years ago
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    Haha well they're my favorite too X-)

  17. anonymous
    • 5 years ago
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    You have to include dt, but yea.

  18. anonymous
    • 5 years ago
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    Oops! Ok! Then I get confused though, where do I go from there?

  19. anonymous
    • 5 years ago
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    Once you take the integral from 0-10 of a rate equation, you've found the total. What I mean by that is this. Let's say you're going to the beach at a rate of 63 miles per hour from the first hour to the fourth hour

  20. bahrom7893
    • 5 years ago
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    let u be -0.2t, du = -.2dt and integrate...

  21. anonymous
    • 5 years ago
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    Yeaaah.... You lost me. I don't understand how to take the integral, the fact that theres an e there messes me up.

  22. anonymous
    • 5 years ago
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    I thought you just put the integral into your calculator. That should be your answer.

  23. anonymous
    • 5 years ago
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    I was just about to explain why that's the answer.

  24. anonymous
    • 5 years ago
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    That seems too easy though, aren't there more steps or something?

  25. anonymous
    • 5 years ago
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    That's it, gimme a sec to explain.

  26. anonymous
    • 5 years ago
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    Ook X-)

  27. anonymous
    • 5 years ago
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    Like I was saying, I drive at 63 miles/hour from the first hour to the fourth (not including). I want the total number of miles driven. Using an integral to do this, I set it up like this:

  28. anonymous
    • 5 years ago
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    \[\int\limits_{1}^{4} (63 miles/hour)dhour\]

  29. anonymous
    • 5 years ago
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    The 1 and 4 are points 1 and 4 on the hour axis (x axis in your calculator)

  30. anonymous
    • 5 years ago
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    Integrating with respect to dhour essentially gets rid of the /hour and returns just the total number of miles driven.

  31. anonymous
    • 5 years ago
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    Go ahead and test it to make sure.

  32. anonymous
    • 5 years ago
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    That integral should return the same number as if you just did the simple multiplying of 3 hours by the 63 miles/hour

  33. anonymous
    • 5 years ago
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    While it's seemingly more work to do it with something simple like my example, it translates exactly the same way as in your example.

  34. anonymous
    • 5 years ago
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    Ok! I think that makes sense! Can you check my answer to my problem though? I'm getting 8647 gallons when I use the calculator.

  35. anonymous
    • 5 years ago
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    If you're given a rate with a time variable, you just take an integral that starts at the first point and ends at the last point and integrate the rate with respect to the time variable, in this case t.

  36. anonymous
    • 5 years ago
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    Okay, lemme load it up

  37. anonymous
    • 5 years ago
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    8646.65

  38. anonymous
    • 5 years ago
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    Close enough :) I don't know why expected that to be so much more complicated, thank you!

  39. anonymous
    • 5 years ago
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    Yea, it's not that bad when you understand what integrals really do.

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