Triple Points

[qwerty123]

Ok, there is a question that is bugging me from the Inv. B '13 test.

It was the one that said:
[unparseable or potentially dangerous latex formula]?

I don't remember the question (don't have the test w/ me), but I'm sure it was something similar.

Anyway I think this problem has something to do w/ combinations such are rearranging the numbers around right?

Is this [unparseable or potentially dangerous latex formula]?

EDIT: it should say non-negative integers

[nsguy1350]

[101dalmatians]

WoW :W
Hey I created the ":W" -> pretty cool right?
Like in art, if u wanna roughly sketch a face then u draw somethin like that right - pointed nose, mouth - getit?

So now on, the coolest symbol is not :O but :W -> or they could tie
who nos? :W

[qwerty123]

@nsguy1350, [unparseable or potentially dangerous latex formula] isn't even one of the answer choices!

However I think I now know my mistake. I believe it is actually [unparseable or potentially dangerous latex formula]

This is because you have to visualize it...

[unparseable or potentially dangerous latex formula]

Now the question is "How many different ways can you arrange the [unparseable or potentially dangerous latex formula]'s"

There are in fact 26 total objects and you are moving 2 things.

[darksaber21]

Here is how I tackled it.

We take one of the numbers and hold it constant. So, lets make p = 0.

So basically, we are left with q + r = 24. Now we can easily figure out the number of combinations for that easily. It is just 25:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]


Now, we change p and make it p = 1. If we repeat the process again. We end up with 24 solutions. And when p = 2, there would be 23.


See a pattern? Basically, we end up with the first 25 numbers in sequence, and as such we basically find the 25th triangular number:
[unparseable or potentially dangerous latex formula]


So nsguy1350 was on the right track using triangular numbers, but it was just the nth number you needed, not the sum of the first nth numbers, and it was the wrong nth number too.

[101dalmatians]

ToO much MaTh :W
ima gonna skip this problem
but is 325 an answer choice? :P

[101dalmatians]

but darksabers method is kinda coOl :O and understandable
@nsguy1350 -> ugh. i seriously dont get wat u did with some casework on there
can u explain?

[nsguy1350]

[darksaber21]

The actual question asked for non-negative solutions. And even if it did ask for positive solutions (where none of the three numbers could equal 0), then the answer is the 22nd triangular number: 253


The answer to that problem was 325, however. 300 wasn't even an answer choice if memory serves me correctly.

[nsguy1350]

[101dalmatians]

that's kinda true nsguy!

[nswiz]

Okay, so im am alway confused on how to do number 18 and number 26 on uil b will someone explain to me how to do it?

[qwerty123]

[qwerty123]

For 18 I noticed that 7-11 = -4

so -7-11 = -18

[nswiz]

Wait how does that work for 18? If you look at UIL D number 18 how would you solve that?. Im not quite understanding your method

[007math]

<redacted> /me is dumb. I apologize.

[007math]

#18 on Inv. D

Plug in what is given and you have [unparseable or potentially dangerous latex formula]. Now plug in the other number:[unparseable or potentially dangerous latex formula]

#26 on Inv. D
This you just have to know through taking calculus or studying all the theorems.

[qwerty123]

[nswiz]

Also, could you help me with number 38 on UIL B.

[101dalmatians]

Well... thetriangle is equilateral
So, since one part of the base is given to be 7cm, the other is 13cm
You know that each of the angles is = 60 degrees in the equilateral triangle ONLY
They ask for the measure of angle PQR
Well
Use the law of cosines now (my favorite :P)

[unparseable or potentially dangerous latex formula]

Solving that, you get [unparseable or potentially dangerous latex formula]

Now, use the law of sines.

[unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] represents the angle RQS

Solving that, you get [unparseable or potentially dangerous latex formula]

You only found RQS
PQR = 180 - RQS [unparseable or potentially dangerous latex formula] (E)

The funny part BTW is that if you do the exact same procedure using 7 instead of 13, you get that PQR = 80 degrees
However, in the picture, the angle looks to be obtuse.
So, 100 is my answer...

[Fredfredburger]

Because of the range restrictions, it is generally better to use the law of cosines to find angles like that.