Question

A rectangular page is to contain 24 cm² of print. The margins at the top and bottom of the page are 1.5 cm and the margins at the other sides of the page are 1 cm. What should be the dimensions’ of the page so that the area of the paper used is minimum?

Answer 1

Let the width of the printed part be ‘x’ cm

Let the height of the printed part be ‘y’ cm

Given, Area of the printed part = 24 cm²

i.e., xy = 24

y = `24/x`

From the given data,

Width of the page

= x + 2(1)

= x + 2 cm

Height of the page

= y + 2(1.5)

= y + 3 cm

∴ Area of the paper

‘A’ = (x + 2)(y + 3)

= `(x + 2)(24/x + 3)`

A = `24 + 3x + 48/x + 6`

`"dA"/("d"x) = 3 - 48`

For maximum or minimum,

`"dA"/("d"x)` = 0

3x² – 48 = 0

x² = 16

x = ± 4  ......[∵ x cannot be negative]

∴ x = 4

Now, `("d"^2"A")/("d"x^2)  = 96/x^3`

At x = 4, `("d"^2"A")/("d"x^2) = 96/x^3`

∴ Area is minimum when x = 4

y = `24/4` = 6

∴ Dimensions of the page:

Width of the page = x + 2 = 4 + 2 = 6 cm

Height of the page = y + 3 = 6 + 3 = 9 cm