A box with a square base, and no top made of a square piece by cutting 5 in. squares from each corner and folding up the sides. The box is to hold 6480 in3. How big a piece of cardboard is needed?

Question

My answer is __ in by __ in.

Thank you so so much!

Answer 1

Let the piece of cardboard be a square of x by x. If you cut a 5 in square out of each corner, then each side of the square base will be x-10. The height of the box, when folded, will be 5. The volume of the box is

V= (height)(width)(length) = 5(x-10)(x-10) = 6480 in³

Expanding this polynomial,

5x²-100x+500 = 6480

5x²-100x-5980 = 0

Dividing through by 5 gives,

x²-20x-1196=0

Using the quadratic equation to evaluate x,

x = 46, -26

Keep the positive one since this is a dimension.

So the dimension of the square cardboard is 46 in by 46 in.

Verifying, 5(46-10)(46-10) = 6480