A box is made from a piece of cardboard

[TONYYEUNG]

Dear All,

Please help me to answer the following question:-

A box is made from a piece of cardboard with dimensions 20cm by 60cm by cutting a square from each corner and folding up the sides. What is the volume of the box of largest volume that can be made this way?

Best Regards,

Tony

 

[galactus]

Let x=the width of the cut-out square.

You then have width 20-2x

length=60-2x

and height x

For \(\displaystyle 0\leq{x}\leq{10}\)

\(\displaystyle \L\\V=(20-2x)(60-2x)x=4x^{3}-160x^{2}+1200x\)

This is the function to be minimized.

 

[soroban]

Hello, Tony!

Galactus solutions is excellent.
\(\displaystyle \;\;\)But did you make a sketch?

      : - - - -  60 - - - - :
    - *-----+---------*-----* -
    : |/////|         |/////| x
    : |/////|         |/////|
    : * - - +         + - - + -
    : |                     |
   20 |                     | 20-2x
    : +                     |
    : * - - +         + - - + -
    : |/////|         |/////|
    : |/////|         |/////| x
    - *-----+---------+-----* -
      :  x  :  60-x   :  x  :

An \(\displaystyle x\)-by-\(\displaystyle x\) square is removed from each corner.

The sides are folded up to form an open-top box.

              *-----------------*
            /                 / | x
          /                 /   *
        /                 /   /
      *-----------------*   / 20-x
    x |                 | /
      *-----------------*
            60-x

Do you see where Galactus got his volume function?

 

[TONYYEUNG]

Thank you for your kind help. I've got a serious problem to handle this new subject and I will ask the questions in here if I have the problem.

Cheers,

Tony