A poster is to have a total area of

180cm2 with 1-cm margins at the bottom and sides and a 2

cmmargin at the top. What dimensions will give the largest

printedarea?

Dimensions of printed area

w = 8.95 cm

h = 13.44 cm

A (max) = 120.28 cm²

Step-by-step explanation:

Lets call " x " and "y" dimensions of the poster area (wide and height respectively). Then

A (t) = 180 cm² = x*y y = 180 / x

And the dimensions of printed area is

A (p) = (x - 2) * (y - 3) then as y = 180/x we make A function of x only so

A (x) = (x - 2) * (180/x - 3) ⇒ A (x) = 180 - 3x - 360/x + 6

A (x) = - 3x - 360 / x + 186

Taking derivatives on both sides of the equation we get:

A' (x) = - 3 + 360 / x²

A' (x) = 0 - 3 + 360 / x² = 0 - 3x² + 360 = 0

x² = 120 ⇒ x = √120 x = 10.95 cm

And y = 180 / 10.95 ⇒ y = 16.44 cm

Then x and y are the dimensions of the poster then according to problem statement

w of printed area is x - 2 = 10.95 - 2 = 8.95 cm

and h of printed area is y - 3 = 16.44 - 3 = 13.44 cm

And the largest printed area is w * h = (8.95) * (13.44)

A (max) = 120.28 cm²