The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volu

Thank you so much melody :))

The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560 cu. units. from a 24 units by 18 units rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made.Show a mathematical solution and illustration.

Let the side lengths of the identical squares be x 

The sides of the box will now be (24-2x), (18-2x) and x

$$\\Volume=x(24-2x)(18-2x)\\\\ 560=x(24-2x)(18-2x)\\\\ 560=4x(12-x)(9-x)\\\\ 140=x(12-x)(9-x)\\\\ 140=x(108-12x-9x+x^2)\\\\ 140=x(x^2 -21x+108)\\\\ 0=x^3 -21x^2+108x-140\\\\ 0=(x-2)(x-5)(x-14)\\\\ x=2\;\;or\;\;5\;\;or\;\;14\\\\ 18-2*14<0 $ so 14 is too big$\\\\\\$$

$$\\If \;\;x=2\\ Volume=2(24-4)(18-4)=2*20*14=560 \;\;good\\\\ If \;\;x=5\\ Volume=5(24-10)(18-10)=5*14*8=560 \;\;good\\\\$$

So the little cut out squares can have side length of either 2 units or 5 units.  

Nice work and graphics, Melody......

Thanks Chris :)