Question 1110305: A small cube is cut from a large cube. The ratio of the remaining volume to the original volume is 19:27. If the small cube has sides of length 14cm, find the length of the sides of the large cube (before the small cube was removed).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the small cube has sides whose length is 14 cm.
the volume of the small cube is therefore 14^3 = 2744 cubic cm.
the ratio of the remaining volume to the original volume is 19:27.
i understand this to mean the volume of the original cube that remains after the volume of the small cube has been removed.
if you let x equal the original volume, then x - 2744 is the remaining volume after the small cube has been removed.
the ratio of the remaining volume to the original volume is 19:27.
this can be written as 19/27.
you get 19/27 = (x - 2744) / x
that'a the ratio of the remaining volume to the original volume.
cross multiply to get 19 * x = 27 * (x - 2744)
simplify to get 19 * x = 27 * x - 27 * 2744
subtract 27 * x from both sides of the equation to get 19 * x - 27 * x = -(27 * 2744)
simplify to get -8 * x = -74088
divide both sides of the equation by -8 to get x = 9261.
that would be the original volume of the cube.
take the cube root of that to get 21.
that would be the length of the sides of the original cube.
the volume of the original cube is 21^3 = 9261.
the small cube is carved out of it, whose volume is 14^3 = 2744.
the volume that remains is 9261 - 2744 = 6517.
the ratio of the remaining volume to the original volume is 19/27.
6517/9261 simplifies to 19/27, confirming the solution is correct.
this simplification is performed by dividing both numerator and denominator by 343.
(6517/343) /(9261/343) = 19/27
your solution is that the length of the sides of the original cube was 21 cm.
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