Question

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

Answer 1

For background knowledge, what math are you in?

Answer 2

AP Calculus

Answer 3

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

Answer 4

Sort of

Answer 5

Do you know what they're used for?

Answer 6

not really

Answer 7

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

Answer 8

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

Answer 9

Do you get that?

Answer 10

Yeah, I think so! Can I just plug the equation into the calculator to graph it and then get my answer from there?

Answer 11

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

Answer 12

I do have one :)

Answer 13

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?

Answer 14

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

Answer 15

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

Answer 16

Haha well they're my favorite too X-)

Answer 17

You have to include dt, but yea.

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

That's it, gimme a sec to explain.

Answer 26

Ook X-)

Answer 27

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:

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

Go ahead and test it to make sure.

Answer 32

That integral should return the same number as if you just did the simple multiplying of 3 hours by the 63 miles/hour

Answer 33

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.

Answer 34

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.

Answer 35

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.

Answer 36

Okay, lemme load it up

Answer 37

8646.65

Answer 38

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

Answer 39

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