There is a rectangular piece of cardboard measuring units by units. Roy cuts out four squares of equal side length, one from each corner of the rectangle, to form a new piece of cardboard which he may then fold up into a rectangular box missing its top. If he cuts out a square of side length 2 units from each corner, then the resulting box has volume 72 cubic units. If instead he cuts out squares of side length 3 units, then the box has volume 42 cubic units. Find the area of the original piece of cardboard.

Expert Answers

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Call the dimensions of the original rectangle a and b, then we are asked to find ab.

After cutting squares of length 2 from all angles, we'll have the box of the height 2 and the horizontal dimensions of a - 4 and b - 4. This way, the volume will be `2 ( a - 4 ) ( b - 4 ) ` and it is given to be 72. This gives us one equation, `( a - 4 ) ( b - 4 ) = 36 .`

The second option is to cut 3 units from each angle. The volume will be `3 ( a - 6 ) ( b - 6 ) = 42 , ` which is the second equation.

Opening parentheses in both equations, we get

`ab - 4 ( a + b ) = 20 ` and `ab - 6 ( a + b ) = -22 .`

It may be considered as a linear system for ab and a+b. We can eliminate a+b by adding the first equation multiplied by 6 and the second multiplied by -4:

2ab = 120+88, ab = 104 (square units).

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