The sum of the perimeters of a circle and square is k where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. 

An open box with a square base is to be made out of a given quantity of sheet of area c². Show that the maximum volume of the box is 

An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminum and folding up the sides. Find the volume of the largest such box.

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off squares from each corner and folding up the flaps. What should be side of the square to be cut off so that the volume of he box is maximum?

319. An open box with a square base is to be made out of a given iron sheet of area 27 square m. Show that the maximum volume of the box is 13.5 cubic m.

A square tank of capacity 250 cubic metres has to be dug out. The cost of the land is Rs.50 per square metre. The cost of digging increases with the depth and for the whole tank is Rs 400 h², where h metres is the depth of the tank. What should be the dimensions of the tank so that the cost be minimum?