SOLUTION: Decreasing cube. Each of the three dimensions of a cube with a volume of y to the third power cubic centimeters is decreased by a whole number of centimeters. If the new volume is

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Question 40998: Decreasing cube. Each of the three dimensions of a cube with a volume of y to the third power cubic centimeters is decreased by a whole number of centimeters. If the new volume is y-cubed - 13 y-squared + 54y - 72 cubic centimeters and the new width is y - 6 centimeters, then what are the new length and height?
Answer by psbhowmick(489) About Me  (Show Source):
You can put this solution on YOUR website!
New volume = y%5E3-13y%5E2%2B54y-72 cc.
Since (y - 6) cm is the new width, so it must be a factor of the new volume.
So, we will write y%5E3-13y%5E2%2B54y-72 in such a way that (y - 6) may be taken common.

y%5E3-13y%5E2%2B54y-72
= y%5E3-6y%5E2-7y%5E2%2B42y%2B12y-72
= y%5E2%28y-6%29-7y%28y-6%29%2B12%28y-6%29
= %28y-6%29%28y%5E2-7y%2B12%29
= %28y-6%29%28y%5E2-3y-4y%2B12%29
= %28y-6%29%28y%28y-3%29-4%28y-3%29%29
= %28y-6%29%28y-3%29%28y-4%29

So clearly (y-3) cm and (y-4) cm are the new length and height respectively.

[N.B.: (y-3) > (y-4); so (y-3) is length being the longest side]