Question

An open box is formed by cutting squares with side lengths of 3 inches from each corner of a square piece of paper. what is a side length of the original paper if the box has a volume of 675 cubic inches?

Answer 1

The volume formula for a square or a rectangle, even, is V=l*w*h
Here, we are given our volume as 675, now we just have to find everything else, right?! Well, if you have a square, all 4 sides are the same exact length, so the length and the width of our formula are going to be the same so we only have to worry about finding a value for one and then use it twice. If you have a square of side length x and you cut 2 squares out of each side measuring 3 inches each, you are cutting away 6 inches. So the side now reflects a length of x - 6. Which is used for the length and the width in our formula. The height of the box will be 3 inches, or in other words, what you cut away from each corner to make a box in the first place! So our formula will look like this: 675 = (x - 6)(x - 6)3. Easiest thing to do is to divide away the 3 to get 225 = (x - 6)(x - 6). Now expand that by FOILing:
$225= x^{2} -12x+36$
Set it equal to 0 to solve for x, the length and width of each side by moving the 225 over to the other side:
$x^{2} -12x-189=0$
Now you just have to factor this.  When you do, you get x values of 21 and -9. But we all know that you cannot have a side length be a negative number so the x value is 21.  That's the side length of the original paper before you cut 6 inches away.