Question

A square piece of cardboard is formed into a box by cutting out 3-inch squares from each of the corners and folding up the sides, as shown in the following figure. If the volume of the box needs to be 126.75 cubic inches, what size square piece of cardboard is needed?

Answer 1

A square piece of cardboard is formed into a box by cutting out 3-inch squares from each of the corners and folding up the sides, as shown in the following figure. If the volume of the box needs to be 126.75 cubic inches, what size square piece of cardboard is needed? : We know the the height of the box will be 3" : The dimensions of base will be (x-6) by (x-6) : The volume equation: Length * width * height = 126.75 : (x-6) * (x-6) * 3 = 126.75 : We can simplify this divide both sides by 3: : (x-6)(x-6) = 42.25 : FOIL (x-6)(x-6) and we have: x^2 - 12x + 36 = 42.25 : x^2 - 12x + 36 - 42.25 = 0 : x^2 - 12x - 6.25 = 0; a quadratic equation : Use the quadratic formula to find x: {{{x = (-(-12) +- sqrt(-12^2 - 4* 1 *-6.25 ))/(2*1) }}} : {{{x = (12 +- sqrt( 144 + 25 ))/(2) }}} You get: {{{x = (12 +- 13)/(2) }}} The positive solution: x = 12.5 : The dimensions of the cardboard square has to be 12.5 by 12.5 : : Check our solution by finding the volume Box dimensions will be 6.5 * 6.5 * 3 = 126 cu/in as given