A cardboard poster containing 32 in^2 of printed region is to have a margin of 2

Question

A cardboard poster containing 32 in^2 of printed region is to have a margin of 2 in at top and the bottom and 4/3in at the sides. determine the dimensions of the smallest piece of cardboard that can be used to make the poster.

Explanation / Answer

If we set the dimensions of the cardboard as : AxB , then The area of the printed region is : S=[A- 8/3][B-4] . Where AB=32 --> B=32/A . Thus, S=[A- 8/3][32/A -4] . To find minimum, we derive S with respect of A, & solve S'=0 : S'=[32/A -4]-(32/A^2)[A- 8/3] S'=0 --> 32/A - 4 - 32/A + 85.33/A^2 =0 32A-4A^2-32A+85.33=0 4A^2=85.33 A^2=21.33 A=4.61 Let's find B : B=32/A = 32/4.61=6.94 The Dimensions are : 4.61 X 6.94

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