Homework SourseAn open box is to be made from a 21 ft by 56 ft rectangular
An open box is to be made from a 21 ft by 56 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.
Round your answer to the nearest integer.
The maximum volume is ______ ft^3.
Round your answer to the nearest integer.
The maximum volume is ______ ft^3.
Solution
Let x be the dimensions of the squares to remove from each corner. Height of sides will be x Length = 56 - 2x Width = 21 - 2x Volume = x (56 - 2x) (21 - 2x) V = 4x^3 - 154x^2 + 1176x . . . V is maximized when V \' = 0 V \' = 12x^2 - 308x + 1176 12x^2 - 308x + 1176 = 0 3x^2 - 77x + 294 = 0 (x - 21) (3x - 14) = 0 x = 14 / 3 ... or ... 21 x = 21 <=== invalid ... would result in a negative width x = 14 / 3 V (x) = 4x^3 - 154x^2 + 1176x V (max) = V (14/3) = 4*(14/3)^3 - 154*(14/3)^2 + 1176*(14/3) V (max) = 68600 / 27 cubic feet about 2540.74 cubic feet
