Question

A sheet of paper must have 18 cm² of printed text, top and bottom margins of 2 cm in height and lateral marsheet gins of 1 cm in width. Determine the dimensions of the entire sheet of paper that minimize its surface area.

Answer 1

Answer:

Let the length of the sheet = x cr

Breadth of the sheet

y cm

Now, area of sheet = 18 cm2

→xy = 18

= y = 1 ... (1)

Now, length of printed portion

= (x – 4) cm

Breadth of printed portion

= (y – 2) cm

Now area of printed portion

= (x - 4) (y - 2)

= A=xy – 2x – 4y + 8

+ A = 18 – 2x ―― 4

4(5) +8

A = 18 – 2x --72 +8

= dĄ = -2 + 72x

For maxima and minima

gives

0

dx

5-2 + 2 = 0

2x2 + 72 = 0

x2 = 36

6

Introduce

-

Now,

d’A

dx?

144

x

d’A

144

3

x=6

(6) 3

dx?

< 0

So, printed area is maximum at x

= 6

Now, from

(1),

Y =

18

18

6

=

3

So, length of the sheet = 6 cm

breadth of the sheet = 3 cm

Regards