The top and bottom margins of a poster are 6 cm and the side margins are each 8 cm. if the area of printed material on the poster is fixed at 382 square centimeters, find the dimensions of the poster with the smallest area.

38.57 cm × 28.93 cm

Step-by-step explanation:

Let x be the length of the poster  ( in cm ) and y be the height ( in cm ) of the poster,

Then the area of the poster,

A = x × y

∵ The top and bottom margins of a poster are 6 cm and the side margins are each 8 cm,

Thus, the total area of the poster other than printed material = 2(6x) + 2(8y) - 4(48)   ( shown in the below diagram )

= 12x + 16y - 192

We have the printed area = 382 cm²

So, the area of the poster = printed area + non printed area

= 382 + 12x + 16y - 192

= 190 + 12x + 16y

∵ Area of the poster, A = xy

⇒ xy = 190 + 12x + 16y

⇒ y(x-16) = 190 + 12x

⇒

Area of the poster,

Differentiating with respect to x,

Again differentiating with respect to x,

For maxima or minima,

y'=0

Since, dimension can not be negative,

If x = 38.57,

A''(x) = positive

Hence, A(x) is minimum,

From equation (1),

Therefore, the required dimensions are 38.57 cm × 28.93 cm

18.8cm hope i at all

step-by-step explanation: