Question

An advertisement consists of a rectangular printed region plus 5-cm margins on the sides and 6-cm margins at top and bottom. If the area of the printed region is to be 238 cm2, find the dimensions of the printed region that minimize the total area. Printed region: l = , w =

Answer 1

Dimensions of printed area

x = 7.58 the length

y = 31.40 cm the height

Explanation:

Printed region P(a) = 238 cm²

Let call x and y dimensions of printed area then:

A = 238 = x*y ⇒ y = 238/x

And the area of the advertisement is

A(a) = L * W where L = x + 6 and W = y + 5

A(x) = (x + 6 ) * ( y + 5 )

Area as a function of x y = 238/ x

A(x) = (x + 6 ) * ( 238/x + 5 )

A(x) = 238 + 5x + 1428/x + 30

A(x) = 268 + 5x + 1428/x

Taking derivatives on both sides of the equation

A´(x) = 5 - 1428/x²

A´(x) = 0 ⇒ 5x² = 1428 ⇒ x² = 57.12

x = 7.58 cm and y = 238/ 7.58 y = 31.40 cm