Question

A rectangular page is to contain 6 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 (in inches) of the page such that the least amount of paper is used. (Round your answers to two decimal places.) smaller value in larger value in

Answer 1

Answer:

A(t)  =  27.86 in²

Dimensions of the paper:

L =  3.73 in

w = 7.47 in

Step-by-step explanation:

The total area of a rectangular page  A  = ( x + 2)* (y + 4 )         x  is the length and  y the wide. x  and  y  are the dimensions of the print area

The print area of the paper is  A =  6 in²       6 = x*y     y  =  6/x

Then print area as a function of x is

A(x)  =  ( x + 2 ) * ( 6/x + 4 )    ⇒  A(x)  =  6  +  4*x  + 12/x + 8

Taking derivatives on both sides of the equation:

A´(x)  =  4  - 12/x²        A´(x)  =  0     4  =  12 /x²

x²  = 12/4       x =√3    x  =  1.73  in  and   y  =  6 / 1.73

y  =  3.47 in

Then the dimensions of the paper  are.

Length  L =  x  + 2  = 3.73 in      and   w  =  3.47 + 4  = 7.47 in

And the least amount of paper is

A(t)  =  3.73* 7.47  =  27.86 in²

To find out if x = 1.73 is an x coordinate for a minimum we get the second derivative

A´´(x)  =  24/x³    is always positive   A´´(x) > 0  then we have a minimum for A at  x = 1.73