Optimization Problem

Question

A newspaper page has top and bottom margins that are 6 cm wide and side margins that are 4 cm wide. If the printed area on the page is 2,400 cm², find the dimensions of the page so that it has the smallest total area.

A. 

42 cm by 74 cm

B. 

52 cm by 68 cm

C. 

40 cm by 60 cm

D. 

48 cm by 72 cm

Answer 1

Let x and y be the dimensions of the printed area.

Printed area = 2400 = xy --> y = 2400/x

Area of the whole page, A = (x+8)(y+12) = (x+8)(2400/x+12) = 12x + 19200/x + 2496

To get the minimum, take the derivative of A and then set that to zero and solve for x.

A'(x) = 12 - 19200/x²

x = √(19200/12) = 40

y = 2400/40 = 60

The dimensions of the whole page are 40+8 = 48 and 60+12 = 72.

The answer is D) 48 cm by 72 cm