# The root numbers in toothpaste (help)

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

cristo
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?

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

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

cristo
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).

Edited... width is 12 cm less than Length

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

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?

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____?

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

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

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.

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...

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

but how do i get 5 from that?

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).

In other words how do i solve for W?

I dont think that 5 is a solution.

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

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

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

ya its three....

good, bye now

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

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