Question

A box is to be made from a rectangular piece of corrugated​ cardboard, where the length is 8 more inches than the​ width, by cutting a square piece 3 inches on each side from each corner and folding up the sides to form a box. the volume of the box is to be 27 cubic inches. find the dimensions of the rectangular piece of cardboard.

Answer 1

The volume of a box like this is found by multiplying the length times the width times the height.  We are told that the length is 8 more inches than the width, so the width is w and the length is w + 8.  If we cut away 3 square inches from each corner, the height when we fold up those corners is going to be 3.  The volume is given as 27, so our formula looks like this:  $27=(3)(w)(w+8)$.  When we do that multiplication, we have  $27=3w^2+24w$.  We need to solve for w so we can then solve for h.  Move the 27 over and set the quadratic equal to 0.  $3w^2+24w-27=0$.  We can then factor out a 3 to make the job easier:  $3(w^2+8w-9)=0$.  Now we can factor to solve for w.  The 2 numbers that add up to 8 and multiply to -9 are 9 and -1.  So (w+9) = 0, (w-1) = 0, or 3 = 0.  Of course 3 doesn't equal 0, so that's out.  w + 9 = 0 so w = -9.  w - 1 = 0 so w = 1.  There are 2 things in math that can never EVER be negative and those are time and distance/length.  So -9 is out.  That means that w = 1.  But don't forget that there was 6 inches cut off each side, so the width is 1 + 3 + 3 which is 7.  The length is w + 8 which means that the length is 7 + 8 or 15.  Those are the dimensions of the rectangle before it was cut.