Consider an axis-aligned cuboid (i.e. with one vertex at the origin and with the 3 edges connected to that vertex along the axis of the 3-dimensional Cartesian coordinate system) with sides of length r, y and z, respectively. 1. Use the method of Lagrange multipliers to find the critical point of the volume of such a cuboid subject to the constraint that it has surface area 6. 2. Find an expression for z in terms of x and y for any configuration which satisfies the surface area constraint. Substitute the expression of z in the function describing the volume of the cuboid to write the problem as an unconstrained two-dimensional optimisation of the resulting function of x and y. 3. Show that the configuration identified in Question 2.1 is a critical point of the function obtained in Question 2.2. 4. Identify a limitation of the approach used in Question 2.2 and deduce about the generality of this approach when compared to the method of Lagrange multipliers.

Consider an axis-aligned cuboid (i.e. with one vertex at the origin and with the 3 edges connected to that vertex along the axis of the 3-dimensional Cartesian coordinate system) with sides of length r, y and z, respectively. 1. Use the method of Lagrange multipliers to find the critical point of the volume of such a cuboid subject to the constraint that it has surface area 6. 2. Find an expression for z in terms of x and y for any configuration which satisfies the surface area constraint. Substitute the expression of z in the function describing the volume of the cuboid to write the problem as an unconstrained two-dimensional optimisation of the resulting function of x and y. 3. Show that the configuration identified in Question 2.1 is a critical point of the function obtained in Question 2.2. 4. Identify a limitation of the approach used in Question 2.2 and deduce about the generality of this approach when compared to the method of Lagrange multipliers.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter9: Surfaces And Solids

Section9.3: Cylinders And Cones

Problem 34E

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