Question

A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

Answer 1

Answer:

 l  = 4  in   d  =  6 in       Note : l represent  the width   and   d represent the height

Step-by-step explanation:

Lets asume the dimensions of a rectangular page is l * d  where l is the wide and d is top-bottom leght

So total area of the page is   A  = l*d       ⇒   d = A/l     ⇒  d = 8/l

Then the print area of the page will be

l = x + 2      (wide)                d  = y + 4    (vertical lenght)

So area of the page is

A(x) = (x + 2 ) * ( y + 4 )      but  y = 8/l    and  l= x +2    ⇒   y = 8 ÷ ( x + 2 )

A (x) = ( x + 2) * ( 8 / x  + 4 )   ⇒ A(x) = 8 + 4x +16/x + 8  ⇒A(x) = 4x + 16 /x

Taken derivative we have:

A´(x)  = 4 + (-1 *(16)/x²   ⇒ A´(x) = 4 - 16/x²

A ´(x) =  0       means      4 - 16 /x² = 0      ⇒ 4 x²  - 16  = 0   x² = 4   x = 2

Therefore   y = 8 ÷ ( x +2)   and y = 2

And the dimensions of the page is

l  = 4  in   d  =  6 in