Question

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?

Answer 1

Dimensions of printed area

w = 8.95 cm

h = 13.44 cm

A(max) = 120.28 cm²

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