# The root numbers in toothpaste (help)

venger
Pardon me for not using latex.

## Homework Statement

A toothpaste box has square ends. The length is 12cm greater than the width. The volume of the box is 135cm^3. What are the dimensions of the box?

## Homework Equations

Quadratic theory, Random pluggin for x, common factoring, Family of functions, graph

## The Attempt at a Solution

Okay, i need 3 numbers when multiplied, will give me 135...
5 into 135 = 27 into 3 times 3 times 3 = 27
okay, good.
and now I'm stuck

## Answers and Replies

Staff Emeritus
The length is 12cm greater than the width.

Can you use this condition to set up an equation for, say, x; the sides of the square end?

venger
Alright im not so stuck anymore, here we go:
V=LWH
135=L(L-12)H
Does this seem right?

Last edited:
robb_
Well, you have two variables and only one equation.
*edit* where is the 15 on the r.h.s. coming from?

Staff Emeritus
Alright im not so stuck anymore, here we go:
V=LWH
135=L(L-15)H
Does this seem right?
No, where does 15 come from?

The simplest way is to take w, say, as the width of the box(note, this is also the height, since the ends are square). Then, the length l, say, is 12 cm greater than w. So, can you express l in terms of w? Then, your formula v=lwh=135 is correct, so substitute into this (again, noting that w=h).

venger
Edited... width is 12 cm less than Length

robb_
okay, now how is the width related to the height?

venger
A toothpaste box has square ends. The length is 12cm greater than the width. The volume of the box is 135cm^3. What are the dimensions of the box?

robb_
The ends are square, which I think you are denoting each side of the square as height and width. Now the two sides of a square are____?

venger
You Smart Little .... Ugh Why am i so Blind.... WOW!!

robb_
I am blind too, just spent more time in this darkness!

robb_
You have the width listed as w. The length is listed as l = w + 12. The height is the other side of the square at the end of the box. What is the relationsip between the width and the height? this will give youone equation with one unknown- no trial and error.

venger
humph....
W H are the same therefore W subs H as W^2
V=(W+12)W^2
135=W^3 +12W^2
0=W^3 +12W^2-135
Cannot go any further therefore must use remainder theorem? Cannot use Quadratic theory, since its not a perfect parabola...

robb_
Now that looks like the right equation! : )
You have a cubic which in general has three roots.

venger
but how do i get 5 from that?

robb_
To find the solution, now you can guess one of the roots by staring at the equation, then it will reduce to a simpler problem, essentially a quadratic (still a cubic though).

venger
In other words how do i solve for W?

robb_
I dont think that 5 is a solution.

robb_
Try a few small integers for w and see if any work.

venger
So... Make the question into an equation then break it down until it is simple to guess the variable?

robb_
I am not sure what you are asking here. For this cubic, guessing a root is probably the simplest method. Have you found it?

venger
ya its three....

robb_
good, bye now

venger
I know I know, I meant i need to use the pluggin theory

robb_
Well, i havent heard of the so called plug in theory but that does work here

venger
I don't know the real name of plug in theory.. but its a theory