A cylindrical vessel of capacity 500 cubic metres open at the top is to be constructed. Find the dimensions of the vessel if the material used is minimum given that the thickness of the material used is 2 cm.

A cylindrical open water tank with a circular base is to be made out of 30 sq metres of metal sheet. Find the dimensions so that it can hold maximum water. (Neglect thickness of sheet).

Young's modulus of the material of a wire is 9.8 x 10^10 N/m^2 and the area of cross-section of the wire is 0.012 cm^2 . Elastic limit of the material of the wire is exceeded when the strain exceeds 1/2400 . Find the maximum load that can be used to stretch the wire without causing a permanent set.

Special purpose coins each weighing 10gms are to be manufactured using two basic metals M_1 and M_2 and a mix of other metals M_3.M_1,M_2"and" M_3 cost₹500,₹800 and ₹800 and ₹50 per gram respectively. The strength of a coin demands that not more than 7 gm. of M_1 and a minimum of 3 gm of M_2 should be used. The amount of M_3 in each coin is maintained at 25% of that of M_1 .Since the demand for that coin is related to its price,formulate the LPP to find the minimum cost of a coin.

Find the quantity of electricity needed to plate a metal of 100 cm^2 area and 10^-2cm thickness using CuSO_4 solution. (density of copper = 9.94 gm//cm^3 and atomic mass = 63.5)