A 27-inch by 72-inch piece of cardboard is used to make an open-top box by removing a square from each corner of the cardboard and folding up the flaps on each side. What size square should be cut from each corner to get a box with the maximum volume? Enter the area of the square and do not include any units in your answer.

Answer:36Step-by-step explanation:Given:Length of the cardboard = 27 inchesWidth of the cardboard = 72 inches.Let "x" be side of the square which is cut in each corner.Now the height of box = "x" inches.Now the length of the box = 27 - 2x and width = 72 - 2xVolume (V) = length × width × heightV = (27 - 2x)(72 - 2x)(x)[tex]V= (1944 -144x -54x + 4x^2)x\\V = (4x^2 - 198x +1944)x\\V = 4x^3 -198x^2 +1944x[/tex]Now let's find the derivativeV' = [tex]12x^2 - 396x + 1944[/tex]Now set the derivative equal to zero and find the critical points. [tex]12x^2 - 396x + 1944[/tex] = 012 ([tex]x^2 - 33x + 162[/tex]) = 0Solving this equation, we getx = 6 and x = 27Here we take x = 6, we ignore x = 27 because we cannot cut 27 inches since the entire length is 27 inches.So, the area of the square = side × side= 6 inches × 6 inchesThe area of the square = 36 square inches.