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Lorene plans to make several open-topped boxes in which to carry plants. She makes the boxes from rectangular sheets of cardboard from which she cuts out 4-in squares from each corner. The length of the original piece of cardboard is 8 in more than the width. If the volume of the box is 1232 in, determine the dimensions of the original piece of cardboard.

Lorene Plans To Make Several Opentopped Boxes In Which To Carry Plants She Makes The Boxes From Rectangular Sheets Of Cardboard From Which She Cuts Out 4in Squa class=

Sagot :

Notice that the height of the box is equal to 4 in. Therefore, the volume of the box is

[tex]V=l\cdot w\cdot h=(x+8-8)\cdot(x-8)\cdot4=4x(x-8)[/tex]

Where l is the length, w is the width, and h is the height of the box.

Therefore, since the volume of the box is 1232 in^2

[tex]\begin{gathered} \Rightarrow1232=4x(x-8) \\ \Rightarrow1232=4x^2-32x \\ \Rightarrow4x^2-32x-1232=0 \\ \Rightarrow x^2-8x-308=0 \end{gathered}[/tex]

Solve the quadratic equation as shown below,

[tex]\begin{gathered} \Rightarrow x=\frac{8\pm\sqrt[]{64+1232}}{2}=\frac{8\pm36}{2}\to\text{ x has to be positive because it is a length} \\ \Rightarrow x=22 \end{gathered}[/tex]

Thus, the answer is that the original length is equal to 30in and the original width is 22in.

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