Question

Calcualate centroid's xbar coordinate of the terahedron (will attach image)
x_0=10, y_0=6 , z_0=15

xbar = ?

Answer 1

I started xbar = 1/v 3 integrals x dv (same for y and z)

Answer 2

then find the normal to the plan ABx AC (i think ?)

Answer 3

because if we find the equation of the plane we can find the inequalities of the reigion then find the volume

Answer 4

malevolence save me :) , haha.

Answer 5

Well, you need to find someway to describe the plane. The easiest way to do this (imo) would be to make a vector from z_0 to x_0 and z_0 to y_0. Cross them. That gives you the normal (call it n). Then you can take n(dot)<x-x_0,y-y_0,z-z_0> =0 Solve for the equation of the plane. From there, the inequalities fall out, i.e., the bounds.

Answer 6

So tell me where you would go with that heading? :P

Answer 7

okay

Answer 8

I would cross (-10,6,0) x (-10,0,15) = 90,150,60 (that doesn't seem right ...)

Answer 9

H/o a second let me look at something.

Answer 10

then I would have 90i + 150j + 60k factor out a 30 => 30(3i+5j+2k) = 30(3,5,2)

Answer 11

Yeah, after that you should get:
3x+5y+2z=30

Answer 12

solve for z then? \[z= (-(3x+5(y-6))/2) \]

Answer 13

becuase we want the upper plane

Answer 14

Exactly:
So:\[0 \le z \le 15-(5/2)y-(3/2)x\]

Answer 15

well I guess probbaly solve for all for the inequalities

Answer 16

Tell me what you get there.

Answer 17

\[0 \le x \le -(5y+2(z-15))/ 3 , 0\le y \le -(3x+2(z-15))/5 \]

Answer 18

Well, remember, after you integrate with respect to z, you don't want any other z's. So what you would do then is take your plane equation:
3x+5y+2z=30
Set z=0. (So, it gives you a line in the x-y plane)
Then solve for x OR y (your choice) I'm going to solve for y.
3x+5y=30
y=6-(3/5)x
So:\[0 \le y \le 6-(3/5)x\].
Then for your x-bounds. Set y=0 (to solve how far in the x direction you need to go.
3x=30
x=10
So:
\[0 \le x \le 10\].

Answer 19

Does that make sense?

Answer 20

ohh!! whoops yeah, brain malfunction.

Answer 21

So your dV would be dzdydx. Because you have to have only y's and x's after you integrate with respect to z. Then only x's when you do y, and your bounds have to be constants when you get to the last part.

Answer 22

that would make it eiaser finding the trip integral haha

Answer 23

Its okay. I'll be back in like 5 minutes, I'm going to eat. Convince yourself that those inequalities work then evaluate it (as best you can) and I'll be right back :P

Answer 24

cool, thanks again. You're a life saver.

Answer 25

so I would do this:
\[v=\int\limits_{0}^{10}\int\limits_{0}^{6-(3/5)x}\int\limits_{0}^{15−(5/2)y−(3/2)x} dzdydx\]

Answer 26

= 150 ( i caved and used wolfram)

Answer 27

now I'm lost as what to do with the volume

Answer 28

Well, this depends, did they give you a density function or are you assuming its constant?

Answer 29

I believe they assume its constant.

Answer 30

Okay. Do you know how to solve that integral though^^?

Answer 31

yeah the triple integral isn't a problem since you explained it in the last problem how to approach these type of problems

Answer 32

oh I think i see what to do!

Answer 33

Whats that? :P

Answer 34

I have it typed out but I'll wait to post it :P

Answer 35

I think to find xbar / ybar etc you do the same thing that we did for volume except if we want xbar we would evalue x dzdydx then put each of the xbar/ybars over the volume to get the centroid. Is that kind of right??

Answer 36

Exactly. This is what I had typed out:
Okay, well from here. You have the volume. And you know xbar is 1/v SSSxdV So take what you have for the volume and rewrite it like this:
\[\int\limits_{0}^{10} \int\limits_{0}^{6-(3/5)x} \int\limits_{0}^{15-(5/2)y-(3/2)x} x dz dy dx\].
Then re-evaluate this:
Your xbar then is the above integral divided by the volume (150).

Answer 37

alright I'm going to do this xbar out just to make sure I do know how to do the trip integral

Answer 38

Okay, tell me what you get BEFORE you divide by 150.

Answer 39

I got 1125/2.

Answer 40

is that what you got for xbar?

Answer 41

Thats what I got for SSSxdV. Then you divide that by 150 (the volume) which gives 3.75

Answer 42

ohhh

Answer 43

H/o I think I dropped something let me double check

Answer 44

i got 375

Answer 45

let me got back I must of messed up

Answer 46

WAIT! 375 is right!!!

Answer 47

I flipped a sign which threw it off ALOT.

Answer 48

ah great success!

Answer 49

Then you divide by 150 giving 2.5!! so xbar=2.5

Answer 50

haha trip integrals are just lots of annoying alegra

Answer 51

let me check it real quick

Answer 52

Agreed^^

Answer 53

100% ! 375/150 = 2.5 thanks again malevolence no way I could of thought clearly through that one

Answer 54

Haha, sorry it was so convoluted. I try to make it concise, its not easy explaining triple integrals via typing xP No problem :) Like I said, here to help xP