Optimization Calc 1

Question

The top and bottom margins of a poster are 2 cm and the side margins are each 3 cm. If the area of printed material on the poster is fixed at 390 square centimeters, find the dimensions of the poster with the smallest area. 

Answer 1

Let w be the width of the printed material and h, the height of the printed material.

The A_PM = wh and

A_P = (w+6)(h+4) (A_P is the total area of the poster and A_PM is the area of the printed material.

We can substitute h = 390/w into the equation we want to minimize:

A_P = (w+6)(390/w + 4) = 390 + 4w + 2340/w +24 = 414 + 4w + 2340/w

Take dA_p/dw and set to 0: 4 - 2340/w² = 0 and w = 24.187 cm and h = 16.125 cm

The second derivative is > 0, therefore, the critical point is a minimum.

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