Question

A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of the equal area from all comers. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.

Answer 1

The sides of the rectangular sheet of paper are in the ratio 8 : 15.

Let the sides of the rectangular sheet of paper be 8k and 15k respectively.

Let x be the side of the square which is removed from the comers of the sheet of paper.

The total area of removed squares is 4x , which is given to be 100.

Now, the length, breadth, and height of the rectangular box are 15k – 2x, 8k – 2x, and x respectively.

Let V be the volume of the box.

Since, volume is maximum when the square of side x = 5 is removed from the corners,

Hence, the lengths of the rectangular sheet are 24 and 45.