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.
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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.
 

15 - 17 Box open at the top in maxima and minima

Problem 15
A box is to be made of a piece of cardboard 9 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made in this way.
 

Problem 16
Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides.
 

Problem 17
Find the depth of the largest box that can be made by cutting equal squares of side x out of the corners of a piece of cardboard of dimensions 6a, 6b, (b ≤ a), and then turning up the sides. To select that value of x which yields a maximum volume, show that

$( \, a + b + \sqrt{a^2 - ab + b^2} \, ) \, \ge \, 3b$