Posted by brianna on .
2 rectangular prisms are proportional. the edges of the smaller prism are half as long as the edges of the larger prism. The volume of the larger prism is 1,024 cubic feet. the volume of the smaller prism is what?
thank you.

math 
Damon,
volume proportional to scale ratio cubed
1/2*1/2*1/2 = 1/8
so
1024/8 = 128 
math 
brianna,
thank you Damon, but how did you get that equation?

math 
Damon,
Well, you see I am a Naval Architect (ship designer)
We test ships in model basins
say the ship is k times as long as the model
then it is k times as deep
and it is k times as wide
so its volume is k * k * k = k^3 times the volume of the model :)
If you imagine the model to be made up of tiny cubes
then the ship would have to have k times as many rows along the length
and k times as many columns in width
and k times as many high
so it would need k^3 times the original number of tiny cubes to fill it . 
math 
brianna,
i know how to get volume,but i need to know how to find the volume of a prism. this prism is proportional to a smaller prism. but the edges of the smaller prism is half as long as the edges of the larger prism. the volume of the larger prism is 1,024 cubic feet. then the volume of the smaller prism is what? i still do not get it. by the way, that's interestinng that you design ships.

math 
brianna,
are you still there?

math 
Damon,
if the small prism is proportional to the large prism
Then the height of the small prism is proportional to the height of the large prism
H = k h
and the length is also proportional
L = k l
and the width is proportional
W = k w
so
L W H = k l * k w * k h = k^3 lwh 
math 
Damon,
That is why I answered your cylinder problem further down the same way. You do not have to know pi.

math 
brianna,
oh, but if the volume of the large prism is 1024, and the edges of the smaller prism is half the length of the larger prism, then what does k mean in your explanation?

math 
Damon,
every edge and other length measurement on the little one is 1/2 that on the big one
so k = 1/2 
math 
Damon,
It does not matter the shape as long as the two objects have the same shape.
The ratio of their volumes is the scale ratio cubed.
(by the way the ratio of any equivalent areas on them is the scale ratio squared) 
math 
brianna,
ohhh, now i get where you got that equation from, but how do i find out what thesize of the edges are, and what does k mean?

math 
Damon,
The volume of the small prism is (1/8) times the volume of the large prism.

math 
Damon,
k is the scale ratio
in this case every length on the small one is 1/2 the equivalent length on the big one
so k = 1/2
and k*k*k = k^3 = 1/8 
math 
Damon,
Now I have to do a pretty long statistics problem for Sherri but will check back later.

math 
brianna,
ok

math 
brianna,
oh, so the length of the small one is 1/8th? but the question says they are rectangular prisms. so they will both have 2 different measurements for the 2 different sixes of the edges

math 
brianna,
Damon,im in 8th grade math, and i don't really get what you are saying. because we haven't really talked about a scale ratio is. ill wait 'till you are done with sherri.

math 
Damon,
OK, I am back from the statistics
Now say we have two cubes
one is 1 cm on every edge
so its volume is 1*1*1 = 1 cm^3
Now another cube (same shape of course)
But it is 2 cm on an edge
so its volume is
2 * 2 * 2 = 8 cm^3
see what happened? 
math 
Damon,
Then what if the second cube were 3 cm on every edge?
volume = 3 * 3 * 3 = 3^3 
math 
brianna,
they are not cubes. they are rectangular prisms. i really sorry if i confused you.

math 
Damon,
It does not matter what shape, only that they are the same shape

math 
Damon,
No matter what shape they are, every height is twice, every width is twice, every length is twice
so every volume is eight times 
math 
Damon,
Now make it harder
Let's say the first cube was 3 cm on an edge
then its volume is
3*3*3 = 27 cm^3
Now say the big one is 6 cm on an edge
its volume is
6 * 6 * 6 = 216 cm^3
BUT I would not do that
I would say
2*3 * 2*3 * 2*3 = 2^3 * 27 = 8 * 27
because my scale ratio is 2
therefore my big volume is 8 times the little one. 
math 
brianna,
no, they are rectangular prisms, not cubes

math 
Damon,
Does not matter shape, as long as the shape is the same. Only the size changes. The volume changes as the cube of every length measurement change ratio.
here is a sphere with 2 cm radius:
vol = (4/3) pi (2^3)
here is a sphere with 4 cm radius (k = 2)
vol = (4/3) pi (4)^3
2^3 = 8
4^3 = 64
SEE  twice the radius> EIGHT times the volume 
math 
brianna,
so wouldn't i divide the volume of the larger prism(1024 cubic feet) by 8

math 
Damon,
YES !

math 
Damon,
If every length on the big one were three times the little one, then the volume of the big one would be 3*3*3 = 27 times the volume of the little one.

math 
brianna,
then i'd divide 1024 and 2, because the smaller one is half as small

math 
Damon,
No, the volume of the little one is 1024 / 8
The smaller volume is 1/8 times the big one 
math 
Damon,
Yes!!
You got it
1024 / 8 = 128 !! 
math 
brianna,
YES!! ithank you sooooo much! i would have gotten ZAP at school if i didn't finish this homework.
THANK YOU SOOOO MUCH!!! 
math 
Damon,
You are welcome  now check that cylinder question down below. Same way.

math 
brianna,
okay,thank you again you are pretty good at this sort of stuff

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