find the dimensions of the poster with the smallest area.

Question

The top and bottom margins of a poster are 8 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 386 square centimeters, find the dimensions of the poster with the smallest area.

Answer 1

Let x is height of poster and y is width of poster. Then aria of poster: A = xy;

But (x + 16)(y + 8) = 386; xy + 8x + 16y + 128 = 386; xy + 16y = 258 - 8x; y = (258 - 8x)/(x + 16).

Then A = x(258 - 8x)/(x + 16); A' = [(258 - 16x)(x + 16) - (258x - 8x²)]/(x + 16)² = (- 8x² - 256x + 4128)/(x + 16)² = 0; x² +32x - 516 = 0; x = (- 16 + √(256 + 516) ≈ 11.785 inch

y = 5.892 inch

Answer 2

For the printed area:

hw = 386, w = 386 / h

For the entire page

A = (h + 16)((386/h) + 8)

Can you simplify, differentiate and answer?