56 - 57 Maxima and minima problems of square box and silo

Problem 56
The base of a covered box is a square. The bottom and back are made of pine, the remainder of oak. If oak is m times as expensive as pine, find the most economical proportion.

Solution:

 

Problem 57
A silo consists of a cylinder surmounted by a hemisphere. If the floor, walls, and roof are equally expensive per unit area, find the most economical proportion.

Solution:

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