How Do You Find the Dimensions of a Prism with a Volume of 42 cm³?

[venger]

Homework Statement

Heres the question I'm tackling:
<b> the height of a prism is x-1, width x-2, length x+3 and the volume is 42cm cubed. Find the dimensions</b>

Homework Equations

Quadratic formula, common factor, family of functions

The Attempt at a Solution

I know that 7, 2 and 3 are multiples of 42 but i do not know how to show my work... I have this so far...
42=(x-1)(x-2)(x+3)
42=(x-1)(x^2-x-6)
42=(x^3-x^2-6x-x^2+x+6)
42=(x^3-2x^2-5x+6)

and I do not know if I am doing things right...

 

[pugfug90]

Is that a rectangular, triangular, or what kind of prism..

 

[venger]

Rectangular prism... sorry for the misleading question

 

[pugfug90]

42=(x-1)(x-2)(x+3)
42=(x-1)(x^2-x-6)

! That should be +x. Makes rest of your calculations wrong.

I got it to be
(x^3)-7x-36=0
or
(x^3)-7x+6=42

And that's all I got so far..
---
PS, 42 cubed? That adds another dyanmic..
Maybe (x^3)-7x+6=(42^3)
?

 

[pugfug90]

HEY, using simple random "pluggin in random numbers", I think the "x" is 4..
Therefore, the sides would be
(4-1), (4-2), (4+3)
or
3, 2, 7

Now, someone else explain an algebraic way of doing this:D

 

[venger]

Ya well I am not really sure how to do all this... but that's a good theory.. It is a rectangular prism and 42 is 42cm^2. Although, I'm not even sure if I'm doing all this right

 

[venger]

Yo ill tinks That works! THANKS

 

[@/@]

You tried to sue any method to solve this equation http://en.wikipedia.org/wiki/Cubic_equations

Two roots of this equation (x^3-7x-36) is imaginary

 

[venger]

Although through the pluggin theory i need to know how to use the right methods to get to 4

 

[turdferguson]

Say you graphed it and that was the only solution

 

[venger]

4 is the correct answer I just do not know how to get to it... My teacher vaguely told me anything about questions like this.

 

[pugfug90]

Graph..
y=(x^3)-7x-36

See where the 0 is..at 4.. What does that mean logically I don't know.:D

 

[cristo]

To solve the equation in this case, you could use the graphical method; i.e. find the roots of the equation x^3-7x-36=0. However, since this is a real problem, looking for a real, positive length, there are certain conditions that the solution must satisfy. Firstly, we are not interested in complex solutions. We note from above that x cannot be negative, or zero, (since then at least one of the sides has negative length, which has no meaning), and x cannot be 1 or 2 (since in each case this gives us a side with zero length)

So, simply start plugging in values starting with 3; you won't have to put many values into obtain the result!

 

[venger]

I don't have a graphing calculator...

 

[@/@]

Some search

you can use the ruffini rule and the rational root theorem

http://en.wikipedia.org/wiki/Ruffini's_rule

look the section
Polynomial root-finding

 

[cristo]

Some search

you can use the ruffini rule and the rational root theorem

http://en.wikipedia.org/wiki/Ruffini's_rule

look the section
Polynomial root-finding

IMHO, it is not necessary to learn these methods of finding roots to a cubic equation for these simple cases. At least one of the roots of problems of this level will be a small number, so "guessing" a solution is enough. Of course, if one wanted to find all the roots, and has found one a, say, using the "trial and error" method, then the equation can be reduced to a quadratic by dividing (x-a) into it. The remaining polynomial can be solved using the usual quadratic formula.