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Advanced MathQ&A Library14· The shape of a holbe of height h andtan α-: hh, show tmaximum if h = (15. A given quantity ofsemi-circular ends. Sthe ratio of the lenm(π + 2). Use Lagr16. Find the volume ofellipsoid x2は x1and (iii) Lagrange mlipler method of optimization of a problem involving-Constraints and n variables.Formulation and Computational Exericsesain the set of necessary conditions for the non-linear programming problem: Maximize2. If f = a3xf + b3xf + ex where x1x2 + x25 + xM" xix-X3, show that the stationary3. Use direct substitution method to minimize f(x, x)+() subject tocylinder is inscribed in a cone of height h. Apply direct substitution method to proveS. Show that the rectangular solid of maximum volume that can be inscribed in a sphere6 Show that, if the perimeter of a triangle is constant, the triangle has maximum area3x5xj subject to the constraintsxx2 3x 2; 5x12x2value off occurs at x," Σα/a, X2 = Σα/b, x,-Σα/c.4. A17. Find the maximumns. Jminat the volume of the cylinder is maximum at height hsis cube. Use direct substitution method.when it is equilateral. Use direct substitution method.multipliers method.18. A meditation centrethe square hall has bof regular pyramid irequired for paintingsubstitution and (ii)7. Show that the diameter of the right circular cylinder of greatest curved surface which can beinscribed 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 cylinderand x = radius of cylinder, S-curved surface of cylinder then maximize S= 2TXh19. A rectangular steel tthe dimensions of thsubstitution (ii) Conssubject toh(r-x) cot a]20. A window is to be dApply constrained variation method to prove that the volume of the biggest right circularcone that can be inscribed in a sphere of given radius is 8/27 times that of the sphere.9. Use constrained variation to maximize the volume of an open cone when the surfacethe perimeter is 40 ft(i) Direct substitutior[Ans. Radius of se80/(+area of the cone is 20π.[Hint: maximize V = 1/3㎡h subject to πrVr2 + h2-20 0]whose surface area is 24.21. Find the volume of tof radius ‘x'. Use (imultipliers method.10.Use constrained variation to maximize the volume of a box made up of thin sheet metalmaximize rvz subject to 2n, + 2)z + 2zr = 24]Question

Asked Mar 27, 2019

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I need you to solve question no 5 only

Step 1

To prove that the volume of a rectangular solid under the given conditions attains its maximum when all the sides are equal (cube)

Step 2

The diagram represents the rectangular box of varying sides L , B and H inscribed inside the sphere of radius R ,say. For varying L,B,H , we obtain different volumes, and we need to show that the maximum volume is attained when all the sides are equal (in other words, the box is a cube)

Step 3

To maximize V (as a function of the vari...

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