Square piece of card with dimensions 24 x 24

Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 24 x 24

In each previous case, the length of the cut-out corner squares has benn 1/6 of the side of the square pieces of card. I predict that for a square piece of card 30 x 30, the side of the small square cut-outs wil be 5 cm.

Square piece of card with dimensions of 30 x 30

Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 30 x 30

Formula to give the maximum open box volume for square shaped pieces of card

x = Side of square shaped piece of card

x/6 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

----------------------------------------------------------------------------------------------------------------------------

Investigation 2 – Rectangular shaped pieces of card

Aim- To find the length of the cut-out corner squares that give the maximum open box volume for rectangular pieces of card of different sizes. The length of the cut-out corner squares will be to 3 significant figures of accuracy.

Method- I will investigate the size of the corner cut-out squares that give the largest open box volume for rectangular shaped pieces of card that have the width to length ratio of 1:2, those being 12 x 24, 24 x 48 and 48 x 96. I will then produce a formula for the maximum open box volume, for all rectangles that have width to length ratio’s of 1:2. I will also investigate the size of the corner cut-out squares that give the largest open box volume for rectangular shaped pieces of card that have the width to length ratio of 1:3, those being 5 x15, 15 x 45 and 45 x 135..I will then produce a formula for the maximum open box volume, for all rectangles that have width to length ratio’s of 1:3. Eventually I will produce a formula that gives the maximum open box volume, for all rectangles. The size of the cut-out corners will be to 2 decimal places of accuracy or 3 significant figures of accuracy.

NOTE – when ‘small side’ is mentioned, it refers to the size of the cut-out corners.

When ‘Length’, ‘Width’ and ‘Height’ are mentioned, they refer to the dimensions of the open box.

When ‘Volume’ is mentioned, it refers to the volume of the open box.

Rows in Italics are those which contain the correct cut-out corner size for the maximum volume of the open box.

Rectangular piece of card with dimensions 12 x 24

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 12 x 24

Rectangular piece of card with dimensions 24 x 48

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 24 x 48

Rectangular piece of card with dimensions 48 x 96

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 48 x 96

In each case above where the ratio of the sides of rectangles is 1:2, the length of small side that has given the largest volume has been the width/4.7

Formula to give the maximum open box volume for rectangular shaped pieces of card with the ratio of 1:2

x = width of rectangular shaped piece of card

x/4.7 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

Rectangular piece of card with dimensions 5 x 15

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 5 x 15

Rectangular piece of card with dimensions 15 x 45

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 15 x 45

Rectangular piece of card with dimensions 45 x 135

Graph comparing the length of Small Side to the Volume for a rectangular shaped piece of card with dimensions 15 x 45

In each of the above cases the length of the small side that gives the maximum open box volume has been width/4.4

Formula to give the maximum open box volume for rectangular shaped pieces of card with the ratio of 1:3

x = width of rectangular shaped piece of card

x/4.4 = side of cut-out corner

Volume = Length x Width x Height

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

________________________________________________________

----------------------------------------------------------------------------------------------------------------------------

Formula to give the maximum open box volume for rectangular shaped pieces of card with the any ratio

x = width of rectangular shaped piece of card

x/ = side of cut-out corner

Volume = Length x Width x Height