Question 280182: 112. Decreasing cube. Each of the three dimensions of a
cube with sides of length s centimeters is decreased by a
whole number of centimeters. The new volume in cubic
centimeters is given by
V(s) = s^3 -13s^2 + 54s- 72.
a) Find V(10).
b) If the new width is s - 6 centimeters, then what are the
new length and height?
c) Find the volume when s = 10 by multiplying the
length, width, and height.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
\ =\ s^3\ -\ 13s^2\ +\ 54s\ -\ 72)
The given function doesn't make any sense.
If you start with a cube that measures  cm on an edge, and you decrease each edge by a whole number  , then each edge would measure  and the volume of the reduced cube would be:
^3\ = s^3\ -\ 3xs^2\ +\ 3x^2s\ -\ x^3)
In order for your given function to make sense in the stated context, then there must be a positive integer that simultaneously satisfies:
Each of these equations has a different solution and none of them are integers. Therefore the given function cannot represent the situation where the measure of the edges are reduced by a whole number amount.
You can determine the first part simply by substitution and a little arithmetic:
\ =\ (10)^3\ -\ 13(10)^2\ +\ 54(10)\ -\ 72)
The second part is trivial. If the width is some value, the length and height must be the identical value so long as the shape remains a cube.
But the last part is nonsensical. Since there is no solution for the integer value of the reduction amount, you cannot compute the actual measure of the edge of the cube based on knowing the original dimension.
And by the way, the problem statement is incorrectly worded because the linear dimensions of a cube are edges. Sides as related to a cube are actually faces which area is measured in square units.
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There is another possibility that just occurred to me. If you re-word the question:
Decreasing cube. Each of the three dimensions of a cube with edges of length s centimeters is decreased by a different whole number of centimeters forming a rectangular prism. The new volume in cubic centimeters is given by
The answer to the first part doesn't change.
The answer to the second part is significantly different. If the width is  and we know that the volume is the product of width, length, and height, if we divide the volume function by the width, we will get a quotient that is an expression equal to the area of one face of the cube bounded by the length and the height.
You can use polynomial long division or synthetic division, but you should get a quotient of:
Which is an expression for the length times the height
Factored:
(s\ -\ 4))
Hence one of those is the length and the other the height. Your choice.
Now if  , the three dimensions are 4, 7, and 6]
And the volume is:
And that should come out to the same value as the first part of the question -- given precisely performed arithmetic.
John
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