Question: You’ve been asked to design the outer structure of a new room, just for New Wave one-hit wonders, for the Rock and Roll Hall of Fame. You have been allotted $77,000 for materials, and are to design the room in the form of a right pyramid (i.e. a pyramid with vertices (0,0,0), (a,0,0) on the x-axis, (0,b,0) on the y-axis, and (0,0,c) on the z-axis, for some positive real numbers a, b, and c). One wall is to be made of glass, costing $3.75 per square foot, and the rest should be poured concrete, costing $9.00 per square foot.

A. Use vectors to compute the total surface area of the room in terms of a, b, and c. Also, determine the volume of the room in terms of a, b, and c.

B. Using Lagrange multipliers, set up the problem which, when solved will reveal the dimensions which give the room the largest possible volume.

C. Solve the problem created in problem 2.