35 - 37 Solved problems in maxima and minima

Problem 35
A page is to contain 24 sq. in. of print. The margins at top and bottom are 1.5 in., at the sides 1 in. Find the most economical dimensions of the page.
 

Problem 36
A Norman window consists of a rectangle surmounted by a semicircle. What shape gives the most light for the given perimeter?
 

Problem 37
Solve Problem 36 above if the semicircle is stained glass admitting only half the normal amount of light.
 

32 - 34 Maxima and minima problems of a rectangle inscribed in a triangle

Problem 32
Find the dimension of the largest rectangular building that can be placed on a right-triangular lot, facing one of the perpendicular sides.
 

Problem 33
A lot has the form of a right triangle, with perpendicular sides 60 and 80 feet long. Find the length and width of the largest rectangular building that can be erected, facing the hypotenuse of the triangle.
 

Problem 34
Solve Problem 33 if the lengths of the perpendicular sides are a, b.
 

29 - 31 Solved problems in maxima and minima

Problem 29
The sum of the length and girth of a container of square cross section is a inches. Find the maximum volume.
 

Problem 30
Find the proportion of the circular cylinder of largest volume that can be inscribed in a given sphere.
 

Problem 31
In Problem 30 above, find the shape of the circular cylinder if its convex surface area is to be a maximum.
 

21 - 24 Solved problems in maxima and minima

Problem 21
Find the rectangle of maximum perimeter inscribed in a given circle.
 

Problem 22
If the hypotenuse of the right triangle is given, show that the area is maximum when the triangle is isosceles.
 

Problem 23
Find the most economical proportions for a covered box of fixed volume whose base is a rectangle with one side three times as long as the other.
 

18 - 20 Rectangular beam in maxima and minima problems

Problem 18
The strength of a rectangular beam is proportional to the breadth and the square of the depth. Find the shape of the largest beam that can be cut from a log of given size.
 

Problem 19
The stiffness of a rectangular beam is proportional to the breadth and the cube of the depth. Find the shape of the stiffest beam that can be cut from a log of given size.
 

Problem 20
Compare for strength and stiffness both edgewise and sidewise thrust, two beams of equal length, 2 inches by 8 inches and the other 4 inches by 6 inches (See Problem 18 and Problem 19 above). Which shape is more often used for floor joist? Why?
 

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

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