14· The shape of a hol be of height h and tan α-: hh, show t maximum if h = ( 15. A given quantity of semi-circular ends. S the ratio of the len m(π + 2). Use Lagr 16. Find the volume of ellipsoid x2は x1 and (iii) Lagrange m lipler method of optimization of a problem involving- Constraints and n variables. Formulation and Computational Exericses ain the set of necessary conditions for the non-linear programming problem: Maximize 3x 5xj subject to the constraintsxx2 3x 2; 5x12x2 2. If f = a3xf + b3xf + ex where x1x2 + x25 + xM" xix-X3, show that the stationary value off occurs at x," Σα/a, X2 = Σα/b, x,-Σα/c. 3. Use direct substitution method to minimize f(x1, X2) = (xL1y+( 4. A 17. Find the maximum ns. Jmin cylinder is inscribed in a cone of height h. Apply direct substitution method to prove at the volume of the cylinder is maximum at height hs multipliers method. S. Show that the rectangular solid of maximum volume that can be inscribed in a sphere 18. A meditation centre is cube. Use direct substitution method. the square hall has b of regular pyramid i required for painting substitution and (ii) 6 Show that, if the perimeter of a triangle is constant, the triangle has maximum area when it is equilateral. Use direct substitution method. 7. Show that the diameter of the right circular cylinder of greatest curved surface which can be inscribed in a given cone is equal to the radius of the cone. Use direct substitution method. [Hint: Let r = radius of cone, α semi-vertical angle of cone, h = height of cylinder and x = radius of cylinder, S-curved surface of cylinder then maximize S= 2TXh subject toh(r-x) cot a] Apply constrained variation method to prove that the volume of the biggest right circular cone that can be inscribed in a sphere of given radius is 8/27 times that of the sphere. 19. A rectangular steel t the dimensions of th substitution (ii) Cons 20. A window is to be d the perimeter is 40 ft (i) Direct substitutior [Ans. Radius of se 80/(+ 9. Use constrained variation to maximize the volume of an open cone when the surface 21. Find the volume of t area of the cone is 20π. 0] Use constrained variation to maximize the volume of a box made up of thin sheet metal whose surface area is 24. of radius ‘x'. Use (i multipliers method. [Hint: maximize V = 1/3㎡h subject to πrVr2 + h2-20 10. maximize rvz subject to 2n, + 2)z + 2zr = 24]