SOLUTION: the volume of a rectangular prism is 96x^3+48x^2+6x. what is its dimensions? using the answer, find an expression that describes the perimeter of one of the prism's faces.

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Question 862089: the volume of a rectangular prism is 96x^3+48x^2+6x. what is its dimensions?
using the answer, find an expression that describes the perimeter of one of the prism's faces.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is an infinite number of possible answers.
Some are easier to write than others.

Maybe x%3E0 and the prism measures 6x by 4x%2B1 by 4x%2B1 .
For that prism, two of the faces will be squares,
measuring 4x%2B1 by 4x%2B1 ,
with a perimeter of 4%284x%2B1%29=16x%2B4 ,
while the other four faces are rectangles measuring 6x by 4x%2B1 ,
each with a perimeter of 2%286x%2B%284x%2B1%29%29=2%2810x%2B1%29=20x%2B2 .

It could be that x%3E0 and the prism measures
3x by 4x%2B1 by 2%284x%2B1%29=8x%2B2 .
In that case there are 3 pairs of faces whose measures are pairs of those dimensions.
Two of the faces will measure 3x by 4x%2B1 ,
with a perimeter of 2%283x%2B%284x%2B1%29%29=2%287x%2B1%29=14x%2B2 .
Another pair of faces measures 4x%2B1 by 8x%2B2 ,
with perimeters of 2%28%284x%2B1%29%2B%288x%2B2%29%29=2%2812x%2B3%29=24x%2B6 .
The remaining pair of faces would measure 3x by 8x%2B2 ,
and would have perimeters equal to 2%283x%2B%288x%2B2%29%29=2%2811x%2B2%29=22x%2B4 .

Maybe the prism is a cube with sides measuring root%283%2C96x%5E3%2B48x%5E2%2B6x%29 ,
and each of its faces has a perimeter of 4root%283%2C96x%5E3%2B48x%5E2%2B6x%29 .

I will not make an infinite list, because I have a finite life, and it's past my bedtime, anyway.