## 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

1. anonymous

For background knowledge, what math are you in?

2. anonymous

AP Calculus

3. anonymous

Okay, that's what I thought. So you know integrals?

4. anonymous

Sort of

5. anonymous

Do you know what they're used for?

6. anonymous

not really

7. anonymous

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

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

Do you get that?

10. anonymous

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

Not really (unless you know have a TI-89 which can integrate)

12. anonymous

I do have one :)

13. anonymous

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

Also, just thought you should know that sunflowers are my dad's favorite so I approve of the name, haha

15. anonymous

$\int\limits_{0}^{10}2000e^(-.2t)$

16. anonymous

Haha well they're my favorite too X-)

17. anonymous

You have to include dt, but yea.

18. anonymous

Oops! Ok! Then I get confused though, where do I go from there?

19. anonymous

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

let u be -0.2t, du = -.2dt and integrate...

21. anonymous

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

I thought you just put the integral into your calculator. That should be your answer.

23. anonymous

I was just about to explain why that's the answer.

24. anonymous

That seems too easy though, aren't there more steps or something?

25. anonymous

That's it, gimme a sec to explain.

26. anonymous

Ook X-)

27. anonymous

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

$\int\limits_{1}^{4} (63 miles/hour)dhour$

29. anonymous

The 1 and 4 are points 1 and 4 on the hour axis (x axis in your calculator)

30. anonymous

Integrating with respect to dhour essentially gets rid of the /hour and returns just the total number of miles driven.

31. anonymous

Go ahead and test it to make sure.

32. anonymous

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

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

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

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

Okay, lemme load it up

37. anonymous

8646.65

38. anonymous

Close enough :) I don't know why expected that to be so much more complicated, thank you!

39. anonymous

Yea, it's not that bad when you understand what integrals really do.