Asked by brianna

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.
Answered by Damon
volume proportional to scale ratio cubed
1/2*1/2*1/2 = 1/8
so
1024/8 = 128
Answered by brianna
thank you Damon, but how did you get that equation?
Answered by 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 .
Answered by 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.
Answered by brianna
are you still there?
Answered by 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
Answered by 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)
Answered by Damon
That is why I answered your cylinder problem further down the same way. You do not have to know pi.
Answered by 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?
Answered by Damon
The volume of the small prism is (1/8) times the volume of the large prism.
Answered by 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?
Answered by 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
Answered by Damon
every edge and other length measurement on the little one is 1/2 that on the big one
so k = 1/2
Answered by Damon
Now I have to do a pretty long statistics problem for Sherri but will check back later.
Answered by 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
Answered by 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.
Answered by 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?
Answered by Damon
Then what if the second cube were 3 cm on every edge?
volume = 3 * 3 * 3 = 3^3
Answered by brianna
they are not cubes. they are rectangular prisms. i really sorry if i confused you.
Answered by 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.
Answered by Damon
It does not matter what shape, only that they are the same shape
Answered by brianna
no, they are rectangular prisms, not cubes
Answered by 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
Answered by 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
Answered by 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.
Answered by brianna
so wouldn't i divide the volume of the larger prism(1024 cubic feet) by 8
Answered by Damon
Yes!!
You got it
1024 / 8 = 128 !!
Answered by brianna
then i'd divide 1024 and 2, because the smaller one is half as small
Answered by brianna
YES!! ithank you sooooo much! i would have gotten ZAP at school if i didn't finish this homework.
THANK YOU SOOOO MUCH!!!
Answered by Damon
No, the volume of the little one is 1024 / 8
The smaller volume is 1/8 times the big one
Answered by Damon
You are welcome - now check that cylinder question down below. Same way.
Answered by Damon
YES !
Answered by brianna
okay,thank you again you are pretty good at this sort of stuff
Answered by omfd xsuayo
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