Calculus issues?

Question

A printed page must contain 60 cm^2 of printed material. There are to be margins of 5 cm on either side and margin of 3 cm on the top & bottom. How long ( width and length) should the printed area be in order to minimize the amount of paper used?

Answer 1

Let the width of the printed area is w .

Let the length of the printed area is l .

The printed area of paper is l*w = 60 square cm .

                                            lw = 60 ---------(1)

The length of the paper is l + 2(3)     (l + 2* 3-cm margins)

                                        = l + 6

The width of the paper is  w + 2(5) 

                                       = w + 10 

The area of the paper is 

A = (l + 6)(w + 10) = lw +10l + 6w + 60 

Eliminate w using (1).

w = 60/l .

Now area of paper 

A =  l( 60/l ) +10l + 6( 60/l ) + 60 

A =  60 +10l + 360/l + 60 

A = 10l + 360/l + 120

To minimize area of paper equate first derivative of area to zero .

A' = 10 -360/l² 

10 -360/l² = 0

360/l² = 10

360 = 10l²

l² = 36

l = ± 6 

Negative value doesn't exist in case lengths.

so l = 6

Put l = 6  in equation (1) 

w = 60/6

w = 10 .

So the dimension of the printed area length = 6 cm  and width = 10 cm .