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A poster consists of a inside printed area framed by a white (unprinted) margin. The top and bottom margins of the poster are each 3 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 96 cm2, find the dimensions of the entire poster with the smallest area. Note that in addition to finding the dimensions, you must also show your justification for WHY these values provide the minimum according to the First Derivative Test for Absolute Extrema.

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Answer:

Step-by-step explanation:

xy=96

y=96/x

A=(x+4)(y+6), using y=96/x we have

A=(x+4)(96/x+6)

A=(x+4)(96+6x)/x

A=(96x+6x^2+384+24x)/x

A=(6x^2+120x+384)/x

dA=(12x^2+120x-6x^2-120x-384)/x

dA=(6x^2-384)/x^2

dA=0 when

6x^2=384

x^2=64

x=8, y=96/x=12

The overall dimensions are x+4 and y+6

A width of 12cm and a height of 18cm (with a minimum area of 216 cm^2)

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