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Question 270176: You have 1 cubic metre bag and wish to reduce the size of the cube so it's volume is reduced by 50%.
What is the formula for this?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula for volume is LWH (length times width times height).
You have LWH = 1 cubic meter.
You want to reduce the volume to 1/2, so you need to divide both sides of this equation by 2 to get:
(LWH)/2 = 1/2
You could divide the length by 2, or you could divide the width by 2, or you could divide the height by 2, and you would get what you want.
If you want to reduce each measurement by the same factor, then you need to determine the appropriate measure to apply to all 3 equally.
let that factor equal to x.
Your formula becomes:
x * L * x * W * x * H = (1/2)*(L * W * H)
This equation is equivalent to:
x^3 * L * W * H = (1/2) * (L * W * H)
Divide both sides of this equation by L * W * H and you get:
x^3 = 1/2
Take the cube root of both sides of this equation and you get:
x = cube root of 1/2 = .793700526
Your common factor is .793700526.
This means that if you multiply each of the dimensions by this common factor, you will get what you want.
An example:
L = 5, W = 7, H = 9
Volume is 5*7*9 = 315
Half of this volume = 315/2 = 157.5
We can get this by:
Taking half the length only to get 2.5*7*9 = 157.5
Taking half the width only to get 5*3.5*9 = 157.5
Taking half the height only to get 5*7*4.5 = 157.5
Reducing each of the dimensions by a factor of the cube root of (1/2) = .793700526 to get
3.96850263 * 5.555903682 * 7.143304734 = 157.5
Any of the 4 ways mentioned will get you what you want which is the volume of the bag reduced by 1/2.
There are other ways as well.
You can change two of the dimensions and leave the third one intact.
You can change any of the dimensions by any amount and then pick up the slack in the third dimension.
An example of that would be:
L = 5
W = 7
H = 9
LWH = 315
half of that equal 157.5
Make the length 6
Make the width 3
Figure out what the height need to be to make the volume equal to 157.5
6*3 = 18
18*H = 157.5
H = 157.5/18 = 8.75
6 * 3 * 8.75 = 157.5
Bottom Line is there there is no one formula that will do it.
You have to specify how you want to do it and then you can derive the formula.
I suspect you want to reduce each of the dimensions by the same amount, which means that the answer of multiplying each of the dimensions by the cube root of (1/2) would be the answer you are looking for.
Assume the bag was in the form of a perfect cube.
Assume each side of the bag was 9 inches in length.
The volume is 9^3 = 729 cubic inches.
Half of that is (1/2) * 729 = 364.5
9 * cube root of (1/2) = 7.143304734^3 = 364.5
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