Length - Side

[Guest]

An open box is to made from a rectangular sheet of cardboard by removing a square from each corner and turning up the sides. Calculate length of a side of each square removed.

Dimensions of cardboard- 9" x 12"
Volume of box- 80 in.^3

[Guest]

eqn for volume of box....

x(9-2x)(12-2x) = 80

This will be a cubic eqn....

solve by trial....
Let x=1
1x7x10 = 70 ...not 80

Let x=2
2x5x8 = 80 ....OK solution

length of side of square is 2"

[Guest]

x = 1.41 inches is also a solution....

from last post (x-2) is a factor of the cubic eqn.
dividing this into the cubic gives the other quadratic factor
solve this quadratic to get x=1.41 or x=7.09
so (x-2)(x-1.41)(x-7.09)=0

[Guest]

Please post solution. Thanks.

[Guest]

A cubic equation has 3 roots, all may not be valid for the problem.....

X=2 gives an exact solution, so x-2 is a factor of the cubic eqn.

also if you divide this factor into the cubic eqn (using synthetic division or long division) you get a quadratic that has 2 roots
if you put this quadratic equal to zero (because if any factor of cubic is zero then the cubic eqn itself is zero) and solve the quadratic gives roots at x=1.41 and x=7.09 approx.
You should be able to get the roots accurately by solving the quadratic accurately...probably use formulae.
the x=7.09 is not valid for the problem but x=1.41 is valid as was x=2

Below is a trial solution to focus in on value of the root near 1.41 rather that use the formulae.
Let x=1.4
1.4x6.2x9.2 = 79.856 too low but root near here.
Let x=1.5
1.5x6x9 = 81 too high so root between 1.4 and 1.5 and nearer to 1.4
Let x=1.41
1.41x6.18x9.18 = 79.993 very near but still low
Let x=1.42
1.42x6.16x9.16 = 80.124 not as near but too high
Let x=1.411
1.411x6.178x9.178 = 80.006 very very near but high
Letx=1.4105
1.4105x6.179x9.179 = 79.9994 near enough for making a box

So the Length of the side is either 2.0 inches or 1.4105 inches

[Guest]

"x(9-2x)(12-2x) = 80"

I can't remember how to do this type of problem. Why is the -2x in both?

[Guest]

Let the side of the square cut out equal "x"
There will be a square cut from each corner.
When the sides are bent up to form a box the depth of the box will be "x"
The width of the box will be 9 minus "x" on each side so that is 9 minus 2x or (9-2x).
Same for length of the box, will be 12 minus 2x or (12-2x).
so the size of the base of the box will be (width by length) (9-2x) by (12-2x).
Volume of box will be depth "x" multiplied by area of the base.
So will be............

(x) multiplied by (9-2x) multiplied by (12-2x)

OR x(9-2x)(12-2x)

and we are told this equals 80

So....x(9-2x)(12-2x) = 80

[Guest]

I remember it now. Thanks again.

[Guest]

One question also:

"x(9-2x)(12-2x) = 80"

Would the FOIL method or another be used to solve ?

[Guest]

FOIL would be used to multiply out the binomials in the brackets....First terms,Outer terms, Inner terms,Last terms. That is onlt the start. Multiplying the two brackets gives you a quadratic, then multiply the quadratic by "x" and you get a cubic equation. It is a cubic equation because we are dealing with volume and terms like x^3.
Then you have to solve the cubic equation for "x".
The numbers are simple so as above we did a trial to see if any easy numbers fitted the equation and found x = 2
If 2 is a solution then x-2 is a factor of the cubic. There are 3 roots for a cubic equation.
Divide x-2 into the cubic to get a quadratic factor and should be no remainder.
Then solve this quadratic to get the other roots x=1.41 and x=7.09.
7.09 is not valid as it is impossible to make a box with sides 7 inches from the 9 x 12 sheet.
The x=2 and x=1.41 are valid.

[Guest]

x(9-2x)(12-2x) = 80
108 - 18x - 24x + 4x^2 = 80
108 - 42x + 4x^2 = 80

How do I proceed from here ?

[Guest]

You have only multiplied out the brackets, you now multiply the LHS by the "x" and re-arrange in standard form.
x(9-2x)(12-2x) = 80
108 - 18x - 24x + 4x^2 = 80 = volume
108 - 42x + 4x^2 = 80
4x^3 - 42x^2 + 108x - 80 = 0

This is a cubic equation, You can build a table and plot a graph, it will have 3 roots ie it will cross the "x" axis in 3 places.
From plotting a graph you can estimate the roots as accurately as you can read the graph. Then use the trial method as on previous posts to get a more accurate value.
We know from the type of problem we are solving that the range of value for "x" will be from 1 to 3 or 4 say otherwise we are making a very narrow small box. So from this it is not hard to get a few trial roots and substitute it into the orig equation.
Take values x=1, x=2, x=3 and you will see that x=2 works out exactly. this gives 1 root at x=2 which satisfies the problem.
To do this you did not even need to multiply out the brackets.

[Guest]

I thought I understood it but I don't.

[Guest]

What do you not understand...

The practical real life problem..
Getting a sheet of cardboard 9 inches by 12 inches and cutting a square out of each corner. then bending the tabs up to form an open box.
1, get a sheet of cardboard 9 x 12 and cut a 2 inch square from each corner, bend the tabs up to form the sides of the box. you should have a box 5 inches wide x 8 inches long x 2 inches deep. It has a volume of 5 x 8 x 2 = 80 cubic inches.
2, get a sheet of cardboard 9 x 12 and cut a 1.41 inch square from each corner, bend the tabs up to form the sides of the box. you should have a box 6.18 inches wide x 9.18 inches long x 1.41 inches deep. It has a volume of 6.18 x 9.18 x 1.41 = 80 cubic inches.
3, Fill one of the boxes with sand or rice or beads etc and empty it into the other box to confirm both have the same volume even though both boxes are different shape.

That is what we are trying to do mathematically....find the size of the square to be cut from each corner of the sheet to leave a box that has a volume of 80 cubic inches. It turns out there are two possible sizes of squares that can be cut.

[Guest]

"108 - 42x + 4x^2 = 80
4x^3 - 42x^2 + 108x - 80 = 0"

I don't understand how to re-arrange the equation.

[Guest]

x(9-2x)(12-2x) = 80 .... this is the equation representing the volume of the box.
It is simply Depth x Width x Length
The depth is "x" because the sides of the squares cut from the corners are "x" inches and when turned up become the sides of the box.
The width is (9-2x) because when the squares are cut from each side of the 9x12 sheet the bit in the middle will be 9 minus 2x inches.
The length is (12-2x) because when the squares are cut from each end of the 9x12 sheet the bit in the middle will be 12 minus 2x inches.
You multiplied out the brackets (Width x Length) to get.....
108 - 18x - 24x + 4x^2 = 80
but you forgot about the "x"...so above line is wrong...
So should be x(108 - 18x - 24x + 4x^2) = 80
and you collected like terms to get....
108 - 42x + 4x^2 = 80
But it should be....
x(108 - 42x + 4x^2) = 80
We still need to multiply by the depth "x" to get rid of brackets..
108x - 42x^2 + 4x^3 = 80
This is an equation representing volume of the box (cubic equation) because it has an x^3 term it is dealing with inches cubed.... 80 is the volume of the box. You could plot this equation on a graph (Y axis for volume and "x" axis for "x") and you would see how the volume changes as "x" changes, and you would see that when x=2 volume=80 and when x=1.41 volume = 80.
To write this equation in standard form normally in order of descending powers L to R and equated to Zero.
When it is written as equated to Zero we have to bring the 80 over to the other side of the equal sign. And when we solve the equation we are solving for roots when Y=0 that is where it crosses the "x" axis. There are 3 roots to a cubic equation.
So continuing from above....
108x - 42x^2 + 4x^3 - 80 = 0
and rearranging in descending powers give....
4x^3 - 42x^2 + 108x - 80 = 0
Now the problem is how do you solve a cubic equation.....
one way is to see if there is a simple root by trial and error...as shown previous posts above...
there are various formulae but these are more complicated....

[Guest]

I understand it now. Thanks for your detailed and thorough explanation.