h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

*stands by with calculator in one hand, pencil in another hand and notepad in a third hand*

A metal sleeve of length 20cm has rectangular cross-section 10cm by 8cm. The metal has uniform thickness, xcm, along the sleeve, and the total volume of metal in the sleeve is 495cm^3.

Derive the equation 16x^2 - 144x + 99 = 0 and solve it to find the value of x.

Whuh? I've drawn myself a picture but I still have no idea what I'm doing.

Also why bother with factorising when you can complete the square instead? and WHY do you want to make all those graphs? What practical worldly thing would I use them for? I hate the fact that school teachers NEVER give you a good reason why you want to learn all these things. You end up knowing lots of theory things on paper but have no idea how to connect them to real life

Because you're learning *how* to learn, and that will stand you in good stead when it comes to something you *will* need to know.

That's what they say, anyway. Not sure I believe it myself.

Factorising/completing the square - if I'm remembering the right things, I found factorising easier than completing the square. I think.

Yeah- I didn't completely "get" completing the square until I was somewhere in high school. Factoring was so much easier for me.

Factoring is important because there are situations in which it can be much quicker to just factor, or to factor and then complete the square, etc. Completing the square is useful, but only for a very set thing -- factoring is useful across a much wider range of situations. Plus, learning calculus would be a if you weren't very comfortable with factoring, I would have to think.


Mikey

Mikeeeeeyyyyy!!!! That didn't help with my homework or show me how to use it in life or what things to use it foooorrrr!!!

OK- first thing you need to do is check for any common factors. If you can't get anything out of that, here's the next step:

1) factor 99. Write down the numbers you can multiply together to get 99. You can tell by the form of the equation that these are probably going to be two negative numbers.

2) Factor 16. Write down those numbers...

3) Figure out which combinations add up to -144.

4) Your answer will take the form:
(ax+b)(cx+d)=0
Then just solve for x. You'll get two answers, but I think you should know which one is the logical one.

Graphing functions is useful to try and determine the behavior of the function for different values without mucking about in calculus. It's also a very nice visual aid for displaying abstract concepts.

Ooh yeah- and the sleeve you're talking about looks kind of like this, right?
______
|------|
| || |
|------|
---------

Well, wouldn't the volume of the metal in the sleeve be the volume of the outside of the sleeve minus the volume contained by the sleeve?

Instead of your problem, here's a similar one --

20x^2 - 73x + 63 = 0

Here is more or less how I taught these problems to my students, although it's insanely impossible to do this all in print. This may be more simplified than you need, but it's hard to know exactly which bits are kerfuddling.

OK, first off, because this equation is ax^2 - bx + x, you know that the factored form will have minuses within both parantheses -- that's the only way you could get a negative 73x and a positive 63, because a negative plus a negative is a negative (-73x), but a negative times a negative is a positive (+63). If they haven't taught you the rules that go with this, let me know, because they are really handy to have.

Anyway, the next step is factoring the outside numbers.

20 63
-----------------------
20, 1 63, 1
10, 2 21, 3
5, 4 9, 7

Next, you figure out what pairing of those numbers could possibly give you -73x -- eventually you'll be able to do this quickly in your head, but it can help to do it methodically at first. You're multiplying the two outside coefficients, and the two inside coefficients, and then adding up the results. For example:

(20x - 63)(x - 1) -- what you get for the middle number there is (20*-1) + (-63*1), which is -83, so it's a nope

(20x - 1)(x - 63) -- you know right away this won't work, because it involves multiplying 20*63, which will give you a number very very very much bigger than 73

Of course, there are a decent number of possible pairings here, so it makes sense to start with the ones that seem the most likely. Here, the fact that the middle coefficient (73) isn't that much larger than the outer coefficients (20 and 63) can be a good clue to start with the pairings of smaller numbers.

(5x - 9)(4x - 7) -- that gives you (5*-7) + (-9*4), so -35 + -36 = -71 -- close, but no cigar

(5x - 7)(4x - 9) = (5*-9) + (-7*4) = -45 + -28 = -73

woo hoo, we have a winner!

So now we know what the factoring is, and we can solve.

(5x - 7)(4x - 9) = 0

For that to happen, one or the other has to be zero.

5x - 7 = 0 so 5x = 7 so x = 7/5
4x - 9 = 0 so 4x = 9 so x = 9/4

Okay yup I got all of that.

I don't understand where the equation for the metal sleeve comes from though.

In my mind, the volume should be

10-2x*8-2x*20

because isn't that the total volume take away the inner space?

I'll look this up at school today and see. I think there's an example with a garden and a pond or something.

Well, what you have the equation for there looks to be just the volume of the empty space, rather than what's left of the metal when you take away the empty space.

Total volume (20*10*8) - empty space [20*(10-2x)*(8-2x)] = metal volume (495)

*sharpens some pencils*

Okay I did work it out in the end. Well, dad and I worked it out. I did it one way, dad did it another and I still maintain that mine is better because you can apply it to any situation whereas you can only use his in that particular problem. I took the space away from the total and he just worked out the sleeve. But I said what if it was a cone or something and then it wouldn't be so neat.

Thanks for the explaining Mikey

It must have been the sharpened pencils really

But still, in what real life situations would I use those graphs please? I can understand the metal sleeve, or the garden pond for the equations, but the graphs?? Oh and mine are always furry, will I get marks taken off?

What sort of graphs are you talking about? I can't seem to tell from the thread.

the ones you get from the equations of (x+3)^2-7 and the like. I understand how you do them but not why.

Do you mean why are they important for school, or for real life? Two very different questions.

For school, they're important because you want to get in the habit of being able to visualize in your head what functions will look like, graphically. For one thing, being able to do this makes it *far* much easier to suddenly spot when your answer isn't near what it should be. On multiple-choice type questions, being able to visualize a function can get you close enough to choose the answer. Being able to visualize functions also makes calculus a lot easier, especially the beginning part.

For life -- well, I do them at work sometimes, is that enough? For example, if we're doing a regression analysis to find the relationship between how many piercings you have and how many books you read this year, we might end up with results that tell us our estimating equation is 4.17x^4 - 12.83x^3 + 2(4.17)x^2 + d = y. That equation then lets me estimate how many books someone has read on the basis of how many piercings they have. If I graph that function, it's very quick and easy to see where the peaks and valleys are, where it's likely that the equation is less reliable (i.e., for the values of x that give me a negative y), and what some of the other driving forces might be.

The same kinds of graphs can be used for all sorts of situations -- I've seen a more complex equation used to create predicted probabilities of cancer over a topographical map in some GIS software, for instance. Luckily, we have computers that do all this graphing for us -- but you need to understand how it works to make sure you're getting what you want out of the computers.

Or was that all more than you really wanted to know?

I like Mikey's way of thinking .

Ah...so I need to understand it really. Thank you for giving me something I can imagine actually doing.

I'll be back with my latest homework to be checked soon. Personally I reckon it's right but you never know. Oh and I've got some daft question about a farmer and his fencing and an equation. I'll bring that to y'all tomorrow.

*charges in!*

Zooooommm!

Right we're onto...er stuff now, quadratic inequalities I believe...

*****************************
x^2-3x+1=0
Find the roots, if any, giving your answer in surd form.
*****************************
I think I've done this one wrong

With the b^2-(4ac) thing I get 9-4=5 so there's 2 roots
Then it goes a bit funny because I don't know what to do with the thing!
If I complete the square I get...
(x-3/2)^2 -9/4=0
x-3/2=Root9/4
x=6/4 +/- Root9/4

so x=6/4 + Root9/4
OR
x= 6/4 - Root9/4

What do we think?

************************
x^2 +2kx+k+6=0
Find values for k for which the equation has equal roots


***********************
***********************
Find the value for p for which the equation has two real roots
x^2 +2px-5p=0


again....
**********************

I have ABSOLUTELY no idea what they are asking me to do
I swear my teacher keeps skipping important things I need to be told. Today she failed to tell us that if x^2 etc etc >0 then all the bits OUTSIDE the graph line were important, instead of the bits inside. Obvious yes, but still it meant that half of us blithely trotted off doing it all wrong.

ok, when it asks you for the values of k that give equal roots, what they're looking for are two different numbers you could pop in for k and get the same roots when you solve. My assumption there is that they're meaning that only one of the two roots would be the same across the two k's.

for a simple example,

(x-4)(x-2) = x^2 - 6x + 8, roots equal 4 and 2
(x-3)(x-2) = x^2 - 5x + 6, roots equal 3 and 2

those don't fit the x^2 + 2kx + k + 6, but that should give you the idea.

for the two real roots, it just means that you need to find the value of p for which the roots you'll get when you complete the square are real numbers, not imaginary ones.

The method I call "plugging and chugging" is a completely acceptable way to go about both of these -- take a small and simple number, plug it in, see if it works. If not, try another small and simple number. As you go, watch for patterns that might give you clues as to which direction to head (i.e., with some equations you'll realize that you need a negative number, or an odd number, or one less than 5...)

As for the first one.... Kat, what's root(9/4)? Simplification is your friend.