(1)A rectangular tank with no top and a volume of 8m3 is to be made from a sheet of metal. The width of the base is 4m. Show that the surface area is given by
S = 4x + 4 + 16/x , where x is the length of the base and find the dimensions of the tank so that minimum material is used.
(2) The breaking weight of a cantilever beam is directly proportional to b*d^2, where b is the breadth of the beam and d is its depth. Find the cross-section dimensions of a beam with a cross-section perimeter of 60cm that has maximum breaking weight.
(3)An oil tank, designed to hold 10 m3, is in the form of a cylinder with hemispherical top.
(i) Show that the surface area (A) of the tank is given by A = 20/r + 5/3 * πr^2 , where r is the radius and h is height of the cylinder.
(ii) Find the dimensions (r and h) of the tank that has least surface area and calculate the least surface area.