Question

Boxes use to ship the components, must have square bases in order or attached square labels to each end. The labels may have standard hazard warnings or handling information. The labels, however, can be magnified or reduced in size, but they must remain square, presently the boxes a base of 10cm square. To pack enough components into a box, the boxes have a capacity of 4000 cm³. The boxes are made of a special fibre-glass reinforced tough plastic to protect the components and the box manufacturer charges £2.50 for an area of 500 cm².

As a cost cutting exercise calculate a box with a square base and a capacity of 4000 cm³ which has the minimum possible area. If 10,000 boxes are use per year by the component manufacturer, how much money will be saved in a year by using the new boxes with the minimum area?

Answer = £10,000

Answer 1

It is assume that box is closed which contains 6 faces

Before modification: height of box = volume of box / base area = 4000/ (10*10) = 40 cm

total surface area of box = total base area + total side area

                                    =2*10*10 + 4*10*40

                                    = 1800 cm²

Modification : let the side of square base = a cm ; height of box = h cm

total surface area, s = 2a² + 4ah

                              = 2a² + 4a*(4000/a²) ............................. (volume = base area * height)

                               = 2a² + 1600/a

for minimum surface area differentiating s w.r.t a

         4a - 1600 / a² = 0

           a³ = 4000

            a = 15.874 cm ; h = 4000/ (15.874*15.874) = 15.874 cm

new surface area after modication = 6* 15.874*15.874 = 1511.9 cm²

Decrease in the area of box after modification = 1800 - 1511.9 = 288.1 cm²

money will be saved in a year = (2.5 / 500)*288.1*10000 = 14405 dollar