Polynomial Word Problem

[asissa]

Here is the problem:
You are designing a marble planter for a city park. You want the length of the planter to be six times the height and the width to be three times the height. The sides should be one foot thick. Because the planter will be on the sidewalk, it does not need a bottom. What should the outer dimensions of the planter be if it is to hold 4 cubic feet of dirt.

The way I see it, the height is x, width is 3x and the length is 6x. But accounting for the one foot thick sides, would it not be x, 3x-2, and 6x-2? If I do it either way, I'm still not getting the answer the book has which is 1x3x6 foot for the sides, or x =1. Obviously, my premise is wrong somewhere. Any clarification to my error would be appreciated.

(x)(3x)(6x)=4
18x^3 - 4 = 0
2 (9x^3 - 2) = 0
9x^3=2 ???

or

(6x-2)(3x-2)(x)
(6x-2)(3x^2-2x)
18x^2-12x-6x^2+4x
12x^2-8x
4x(3x-2)
4x=0
3x=2????

Thanks -

 

[tkhunny]

(x)(3x)(6x)=4
18x^3 - 4 = 0
2 (9x^3 - 2) = 0
9x^3=2 ???

No good. Your second attempt is the right one.

18x^2-12x-6x^2+4x

This is where you wandered off. You should have gotten 18x^3 - 18x^2 + 4x = 0

Be more careful and give it another go.

 

[asissa]

Thank you - I see my mistake, but I still have a question.

Let's start with:
(6x-2)(3x-2)(x)=4
(6x-2)(3x^2-2x)=4
19x^3 - 12x^2 - 6x^2 + 4x =4
18x^3 - 18x^2 + 4x -4 = 0
2(9x^3 - 9x^2 + 2x - 2)
9x^2(x-1) + 2(x-1)
(9x^2 + 2)(x-1)
given you x = undefined and x=1.

My question is this. Height = 1. I guess I plug into the basic (6x and 3x) to get the sides and width and ignore the 1' foot sides that I had to take into account to get the volume. This is slightly confusion, as in the book example they show plugging the answer for 1 in to the equation for that side (which would give me 1 x 1 x 4, not the answer 1 x 3 x 6). I guess the plain dimensions would not have to take into account the thickness of the sides. Trick question!! Thanks for you help.

 

[tkhunny]

Not a trick question. Look at your definitions.

"the height is x, width is 3x and the length is 6x."

You should have stated explicitly that these are the OUTER dimensions. That is what you have done. You just didn't say it.

Given the OUTER dimensions of 1 x 3 x 6, you get INNER dimensions of (1 - 0) x (3 - 2) x (6 - 2) or 1 x 1 x 4 - exactly what was desired.