Question

Lorene plans to make several open-topped boxes in which to carry plants. She makes the boxes from rectangular sheets ofcardboard from which she cuts out 2-in squares from eachPLEASE CHECK PHOTO

Answer 1

Given:

When the cardboard is folded to become a box (cuboid), it will have the following dimensions after the cut of squares from each corner;

`\begin{gathered} l=(x+4)-2-2=x+4-4=x \\ w=x-2-2=x-4 \\ h=2 \\ \\ The\text{ volume }V=792in^3 \end{gathered}`

The volume of a cuboid is given by;

`\begin{gathered} V=lwh \\ 792=(x)(x-4)(2) \\ Dividing\text{ both sides by 2;} \\ (792)/(2)=x(x-4) \\ 396=x^2-4x \\ \\ Collecting\text{ all sides to one side to form a quadratic equation;} \\ 0=x^2-4x-396 \\ x^2-4x-396=0 \end{gathered}`

Solve the quadratic equation by factorization;

`\begin{gathered} x^2-4x-396=0 \\ x^2+18x-22x-396=0 \\ x(x+18)-22(x+18)=0 \\ (x-22)(x+18)=0 \\ x=22,x=-18 \\ \\ Since\text{ the dimension of a box can not be negative, then;} \\ x=22in \end{gathered}`

Hence, the dimension of the original piece of cardboard is;

`\begin{gathered} (x+4)\text{ by }x \\ \\ Substitute\text{ the value of x, the dimension of the cardboard is;} \\ (22+4)\text{ by }22 \\ 26in\text{ by }22in \end{gathered}`

Therefore, the dimensions of the original piece of cardboard are 26 in by 22in