A4 Paper to Container

Question

What is the largest
container with
rectangular sides (and
no lid) that can be
made from a sheet of
A4 paper?

Answer 1

A4 paper  measures 210 × 297 millimeters or 8.27 × 11.69 inches 

 

The total surface area of A4 paper=62370 square  millimeters or 96.6763 square inches 

 

The largest container with rectangular sides (and no lid) that can be made from a sheet of A4 paper will have 5 surfaces with a width of l millimeters or L inches with 1 surfaces having dimensions (210-2l) ×  (297-2l)  millimeters or ( 8.27- 2L) ×(11.69-2L)  inches and 2 surfaces with (210-2l) ×  l millimeters dimensions or  inches dimensions and another 2 surfaces with (297-2l) × l millimeters dimensions or (11.69-2L) × L inches dimensions and 4 pieces of l × l millimeters dimensions or L× L inches dimensions will be cu( 8.27- 2l) × Lt off from the A4 paper

 

Thus the total surface area of A4 paper used to make a container with rectangular sides (and no lid) can be summed up as follows:

(210-2l) × (297-2l)+ 2 (210-2l) × l + 2 (297-2l) × l + 4 l × l =62370 ------ (1)

OR

( 8.27- 2L) ×(11.69-2L)+ 2 ( 8.27- 2l) × L + 2 (11.69-2L) × L + 4  L× L =96.6763 ----- (2)

 

Solving (1)

62370-420l- 594l +4l² + 420-4l+594-4l +4l² =62370

=>8l² - 1022l+1014=0 .............. subtracting 62370 from both sides and adjusting coefficients and operators

=> 8l² -8l -1014l + 1014 = 0 ......................... adjusting coefficients and operators

=>8l(l-1) -1014 (l-1) = 0 ......................adjusting coefficients and operators

=>(l-1)(8l-1014)=0 .......................adjusting coefficients and operators

Hence, either l-1=o or 8l-1014=0

                =>  l=1 or         8l =1014 .............. subtracting 1014 from both sides

                            or           l= 126.75 .............. dividing both sides by 8

thus the possible dimensions of the container is either (210-2) × (297-2) × 1 millimeters = 208 × 295 × 1 millimeters

or (210-253.5) × (297-253.5) × 126.75 millimeters which is physically impossible as in reality a container cannot have a negative dimension

 

Solving (2)

96.6763-16.54L-23.38L+4L²+ 16.54-4L+23.38-4L + 4L² =96.6763

=> 8L²-8L-39.92L+39.92=0  .............. subtracting 96.6763 from both sides and adjusting coefficients and operators

=>8L(L-1)-39.92(L-1)=0 ......................... adjusting coefficients and operators

=>(L-1)(8L-39.92)=0 ......................... adjusting coefficients and operators

Hence, either L-1=o or 8L-39.92=0
              =>    L=1 or 8L =39.92 .............. subtracting 39.92 from both sides
                            or     l= 4.99 .............. dividing both sides by 8
thus the possible dimensions of the container is either (8.27-2) × (11.69-2) × 1 inches = 6.27 × 9.69 × 1  inches
or (8.27-9.98) × (11.69-9.98) × 4.99 inches which is physically impossible as in reality a container cannot have a negative dimension

 

 

Therefore, the largest container with rectangular sides (and no lid) that can be made from a sheet of A4 paper may have dimensions 208 × 295 × 1 millimeters or 6.27 × 9.69 × 1 inches