I need help with optimizing a minimum

Question

A rectangular page is to contain 30 square inches of printable area. The margins at the top and bottom of the page are each 1 inch, one side margin is 1 inch, and the other side margin is 2 inches. What should the dimensions of the page be so that the least amount of paper is used?

Answer 1

x = length of horizontal side of printable area

y = length of vertical side of printable area

 

xy = 30 in^2

 

y = 30/x

 

page height = y + 1 + 1

page width = x + 1 + 2

 

A = area of entire sheet of paper

 

A = (x+3)(y+2)

 

A = (x+3)(30/x+2)

 

A = x(30/x+2)+3(30/x+2)

 

A = 30+2x+90/x+6

 

A = 2x+90/x+36

 

Extrema occur when A' = 0:

 

A' = 0 = 2 – 90/x^2

 

1 = 45/x^2

 

x^2 = 9*5

 

x = 3√(5), x has to be positive

 

y = 30/x = 30/(3√(5)) = 10√(5)/5 = 2√(5)

 

What should the dimensions of the page be so that the least amount of paper is used?

 

page width = x + 3 = 3√(5) + 3 = 3(√(5) + 1)

page height = y + 2 = 2√(5) + 2 = 2(√(5) + 1)
So page is in landscape mode with width : height :: 3 : 2.

 

Interesting side note: √(5) + 1 = 2*(Golden Ratio)

 

Fun exercise:

 

Calculate (√(5) + 1)/2 and save it in your calculator's memory.

 

Clear the calculator (but not the memory) and key into your calculator ANY positive number.

 

Repeat the following sequence until the resulting number is the same as the previous result:

 

1/x + 1 =

 

Compare your final number with the Golden Ratio you stored in memory.