Posted by **Andrew** on Monday, July 18, 2011 at 1:42pm.

an open box is made by cutting out squares from the corners of a rectangular piece of cardboard and then turning up the sides. If the piece of cardboard is 12 cm by 24 cm, what are the dimensions of the box that has the largest volume made in this way?

- Calculus -
**Reiny**, Monday, July 18, 2011 at 2:17pm
Let each of the cut-outs be x

then the base of the box is 24 - 2x

the width of the box is 12-2x

and its height is x

Volume = V = x(24-2x)(12-2x)

expand, differentiate and set equal to zero

you will be solving a quadratic.

- Calculus -
**Andrew**, Monday, July 18, 2011 at 2:48pm
Would that make the dimensions x=6 and 12? or did I do that wrong?

- Calculus -
**Reiny**, Monday, July 18, 2011 at 3:04pm
How did you possible get these answers ?

They make no sense.

If x = 6, then you would be cutting away 12 cm from the base of width 12, so nothing is left.

x = 12 would be physically impossible, you would be cutting away 24 cm from the 12 cm width !

Did you even differentiate ?

Show me what you did.

- Calculus -
**Andrew**, Monday, July 18, 2011 at 3:09pm
i got V= 4x^3-72x^2+288x

V=12x^2-144x+288=0

Then I foiled that and my answers were x=6 and x=12. Clearly I have no idea what I'm doing, I have a test in my class on wednesday and I need to be able to make sense of this in my homework due tomorrow

- Calculus -
**Reiny**, Monday, July 18, 2011 at 3:26pm
Your equation 12x^2-144x+288=0 is correct

from there, divide each term by 12

x^2 - 12 + 24 = 0

This does not factor (FOIL is used to expand, not in factoring)

so ...

x = (12 ± √(144 - 4(1)(24))/2

= (12 ± √48)/2

= (12 ± 4√3)/2 = 6 ± 2√3

= aprr. 2.536 or 9.464

but clearly x < 6 or else the width would become negative, so

x = 2.536

length = 24-2(2.536) = 19.928

width = 12 - 2(2.536) = 6.928

height = 2.536

- Calculus -
**Onisha**, Wednesday, September 17, 2014 at 7:40pm
x(24-2x)(18-2x)

(24x-2x^2)(18-2x)

4x^3-84x^2+432

4x(x^2-21x+108)

4x(x-12)(x-9)

x=12,9,0

HELP!

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