HELP!! Water flows into a tank according to the rate F(t) = (6+t) / (1+t) and at the same time empties out at the rate E(t) = ln(t+2) / (t+1), with both F(t) and E(t) measured in gallons per minute. How much water, to the nearest gallon, is in the tank at time t = 10 minutes. You must show your setup but can use your calculator for all evaluations.

Question

welshfella · Answer

net flow = F(t) - E(t)

erinkb99 · Answer

Do I just plug in 10 then for both and then subtract? That seems too easy.

welshfella · Answer

yes
that would only give you the rate of flow at time t = 10 minutes
the rate is not constant It varies with time.

erinkb99 · Answer

Wow, ok. I'm gonna post a few more questions. Can you hep with those?

parthkohli · Answer

No, I believe there's integration involved.

welshfella · Answer

ive havent answered this one  but i;ve just been called away...

erinkb99 · Answer

@ParthKohli that would make sense. Can you explain?

parthkohli · Answer

Since the problem talks about gallons per minute, i.e., rate instead of just gallons, something tells me that those functions given to you are rates of change. 

I could be very wrong though - I have no trust in people who frame word problems these days. Haha

erinkb99 · Answer

I think you're right lol

parthkohli · Answer

So I believe $F(10) - E(10)$ would only tell you the rate of change of volume at the instant $t = 10 min$. 

For finding the absolute volume in the tank, the expression turns out to be$$V(10) = \int_0^{10}(F(t)-E(t)) dt$$

erinkb99 · Answer

Ok so I have to find the anti-derivative of each then?

parthkohli · Answer

Yes indeed... that's the only way to do it.

erinkb99 · Answer

For F(t) its 6*ln(1+t) + t - 1*ln(1+t) right? I'm stuck with E(t)

parthkohli · Answer

Yeah E(t) isn't gonna be elementary damn

parthkohli · Answer

If you're in high school, this problem is not for you. Abort mission.

erinkb99 · Answer

Yea, I'm in high school. This is one of my assignment problems.

parthkohli · Answer

Did these guys not think the problem through?!

You'll just have to hope that the problem doesn't want you to integrate them then

parthkohli · Answer

Because many (in fact most) functions' antiderivatives cannot be expressed in an elementary form, if you know what I'm talking about.

erinkb99 · Answer

I do. Oh well. Can you help with a few more?

erinkb99 · Answer

They are less complicated haha

parthkohli · Answer

Yeah sure

parthkohli · Answer

But you need to report this one problem to your instructor.

erinkb99 · Answer

I will. Thanks.