"Find the volume of the largest open box.."

• Apr 7th 2010, 05:11 PM
Wolvenmoon
"Find the volume of the largest open box.."
Here's the problem:

"Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

I have no provided diagram, the answer is 1024 cubic inches.

Here's what I've done:

V=x*y*z

y = z, y*z = 24.

y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.
• Apr 7th 2010, 05:31 PM
skeeter
Quote:

Originally Posted by Wolvenmoon
Here's the problem:

"Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

I have no provided diagram, the answer is 1024 cubic inches.

Here's what I've done:

V=x*y*z

y = z, y*z = 24.

y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.

let $x$ = side length of squares cut from each corner

$H = x$

$L = W = 24 - 2x$

$V = LWH = (24-2x)^2 \cdot x$

$V = (576 - 96x + 4x^2) \cdot x$

$V = 576x - 96x^2 + 4x^3$

find $\frac{dV}{dx}$ and maximize
• Apr 7th 2010, 05:37 PM
Quote:

Originally Posted by Wolvenmoon
Here's the problem:

"Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides."

I have no provided diagram, the answer is 1024 cubic inches.

Here's what I've done:

V=x*y*z

y = z, y*z = 24.

y=z=square root of 24 - 2x, substitute into the equation for volume, differentiate, set v'(x) = 0, solve, figure out which one doesn't break my domain, substitute into the original equation...and it doesn't work.

Hi Wolvenmoon,

as the same size square is being cut from all 4 corners,
simply label the side of that removed square "x".

Then, since 2 squares are being removed from any side,
the base of the box is a square of sides 24-2x.
The height of the box is x.

The box volume of the bos is (base area)(height)=x(24-2x)(24-2x).

You can either multiply this out and differentiate wrt x
or use the product rule.

$\frac{dV}{dx}=0$

$\frac{d}{dx}\left[x(24-2x)^2\right]=0$

$2(24-2x)(-2)x+(24-2x)^2=0$

$-4x(24-2x)+(24-2x)(24-2x)=0$

$4x=24-2x$

$6x=24\ \Rightarrow\ x=4$

$V=4(24-8)^2=4(16^2)=4(256)=1024$