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.

Solution:

 

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.

Solution:

 

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

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

Solution:

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