The Open Box Problem

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The Open Box Problem 

An open box is to be made from a sheet of card as shown below. The corner squares are to be cut-off.

The card is then folded along the dotted lines to make the box.

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Investigation 1 – Square shaped pieces of card

Aim – To find the length of the cut-out corners that gives the maximum volume for the open box formed for any sized piece of square card. The length of the square cut will be to 3 significant figures of accuracy.

Method – I will investigate what length of cut-out corners will give the largest volume ofr square pieces of card with dimensions 12 x 12, 18 x 18, 24 x 24 and 30 x 30.

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.

Square piece of card with dimensions 12 x 12

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

Square piece of card with dimensions 18 x 18


 


Graph comparing the length of Small Side to the Volume for a square shaped piece of card with dimensions 18 x 18

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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 ...

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