Optimization Problem, Calculus 1

Question

The top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. If the area of printed material on the poster is fixed at 1,536 cm², find the dimensions (in cm) of the poster with the smallest area.

Answer 1

Let x be the length and y be the width of the printed area.

The area of the printed area is:

xy = 1536 cm² --> y = 1536/x

The area of the entire poster is:

A = (x+24)(y+16)

A(x) = (x+24)(1536/x+16)

To minimize the area of the poster

A'(x) = (x+24)(-1536/x²) + (1536/x+16) = (16x²-36864)/x²

Set the numerator to zero and solve for x

16x²-36864 = 0

x = ±√2304 = ± 48 Use +48 since this is a dimension

y = 1536/48 = 32

The length of the entire poster is x+24 = 72 cm

The width of the entire poster is y+16 = 48 cm

Use 2nd derivative test to make sure this is the minimum

A''(48) = 2/3 >0 So this is a minimum