Algebra & Calculus

Hi guys, haven't been on YoungWriters for a while now, but I thought I'd pop in and ask for some help.

Spoiler

http://i1106.photobucket.com/albums/h38 ... i/math.jpg


The image is too big, sorry about that.

Basically I'm pretty bad with my physics and math, and I'm struggling with these questions.

Any physicists/mathematicians out there?

In exchange I'll review whatever of your work you'd like me to take a look at, or I'll make you a signature (that you can't use on YWS, lol, seeing as our sigs don't support images, but it'd still be cool? ).

Thanks, jhoijhoi

Tank draining problems! <3 I did a ton of these in school as a biochemical engineer.

It's reeeeeeeeeeeeeeally late right now and I need to sleep, but I did the firstest first part of part 3 since it looked fun. I'll continue more at a more reasonable hour tomorrow.

I think I did a notation error in the very bottom... it doesn't look like a differential equation, lol. I'll figure it out later, again when it's not crazy late. Still, the set up should be good.

Hiya Snoink Thanks for the starter, that does indeed help me out It's so much easier to see what's happening in the question when it's explained visually! Looking forward to any further help, but you're not obliged to if you're busy ^^

OH. I get it. (You know it's late because I am slow!)

Anyway, here's the differential equation (ODE) for volume. I'll do the rest later, lol.

By the way, in the second step, you're taking the derivative of everything. That's why the constant disappears. This is not simple division!

How are you so smart? I didn't even think of doing it that way, I was so confused :/

I did graduate with a degree in Biochemical Engineering! These problems, to me, are fun.

Also: holy crap, lol. That tank draining problem. Have you dealt with nonhomogeneous ordinary differential equations yet? Because, from what I have done for the concentration so far, the problem goes to a nonhomogeneous answer, but it seems a bit advanced for a calculus class. Or maybe you Aussies are smarter than us?

In any case, I can still help you with it. I aaaaaaaalmost have the answer, after all! I just need to write it out and I don't want to confuse you entirely. >.>

Let's tackle the limit problem.

At first glance the problem looks ugly. However, with some simplifying, the problem breaks down.

The first leap is to rationalize the radical in the denominator. That gets rid of the square root on the bottom, turning it into a nice simple t-4 term.

The second leap is a pretty big one. You have to factor the (t^3-64) term. This requires using the very obscure formula for factoring the sum of cubes. It is something from algebra and is rather obscure. I only barely remember it because I've seen it on the wall of the tutoring center I volunteer at.

After that though, we find a t-4 term on top to cancel with the one on the bottom. From there, we can simply plug in t=4 to get the answer.

Thank you both for your help. I don't know why I can't do these questions myself.

As always with these kind of homework sheets, they expect us to make "leaps" of faith and apply past knowledge to "new" questions. I have this problem where I can't do an equation unless I've seen something similar before :/

So while I can do simple limits, I need to be refreshed to see all the rules implemented, so thank you, Griffin

And nah, we're not smarter, if I was smarter I wouldn't be asking for help lol. It's a second year university calculus class ^^

Your parents must be proud, the two of you being so brainy and all.

The concentration thing is pretty nasty, so warning. But it's not that bad.

So, first of all... the first approach is to NOT do it by concentration. Seriously. Save that for the last step. Instead, you're going to be doing a mass balance. Basically, mass goes in, mass goes out. In units of [g/min]. You will not attempt to do the concentration until AFTER figuring out this using the mass. Seeing that C = m/V (Concentration = mass/Volume] it's going to be okay.

But really, the reason why you don't want to do C at first is because it's ridiculously hard. Since the Volume is NOT constant, then you have another factor to deal with. Just divide m with V afterwards. -_-

Anyway...

Step #1:

This is basically where we set up the expression (differential = rate in - rate out) and change around the variables to make it look pretty. This looking pretty thing is pretty much necessary.

http://i793.photobucket.com/albums/yy21 ... G_1156.jpg

Step #2:

This is where we use the concept of integrating factors to clean up the problem. It looks nasty, but it makes our lives wonderful and beautiful.

http://i793.photobucket.com/albums/yy21 ... G_1158.jpg

Step #3:

If you stick the integrating factors in the original expression and use the product rule, a much, much, much more simplified expression comes out.

http://i793.photobucket.com/albums/yy21 ... G_1162.jpg

Step #4:

Hey, look it! If you integrate both sides, you can solve for m! Sweeeeeeeeeeeet!

http://i793.photobucket.com/albums/yy21 ... G_1167.jpg

So that's all I did. Anyway, what you would do next is to divide m by V. Then you would find the value of that arbitrary constant by finding C(0) = 0.1 g/L at t = 0.

Then you can do the rest of the problems easy because it's just plug and chug by then. The hard part is formulating the equation.

Also, you'll note that I didn't simplify the constants into numbers. I typically like putting in the constants at the end. Otherwise, I typically mess up the units. Make sure you're careful with this!

o.O

Woah, that is a whole lot of steps. How am I supposed to remember that all? I will give it a go, trying all the steps you did, and see if I can "work out" exactly the frame of mind you had when you did it so I can grasp the steps completely.

Thank you so much for your help, Snoink

Really quick sig, I can make a more detailed one if you'd prefer. Just thought I'd show my thanks ^^ I'll make one for Griff too, a bit later.

So cute!!! <3

haha






Pigs are awesome focal points ^^

And for Griff

Thanks!

Also! If it makes you feel better... we spent several hours just reading books about differential equations and calculus to figure out how to work out some of the problems. Plus, we used several computational programs that we have (he's an applied mathematician, I am a biochemical engineer, so we have quite a lot of them at our disposal!) to figure out whether we were on target or not. Since we've both graduated and we never have to do those problems again for school, it's fun for us. Besides, we've dealt with those problems in the past too. However, I can imagine that when you encountered these problems for the first time, you were a little overwhelmed. I had dreams about the concentration problem. XD