Question

you are designing a rectangular poster to contain 50 in.² of printing with a 2 - in. margin at the top and bottom and a 1 - in. margin at each side. what overall dimensions will minimize the amount of paper used? what is the vertical height of the poster that will minimize the amount of paper used? h = 14 in. what is the horizontal width of the poster that will minimize the amount of paper used? w = □ in.

Explanation:

Step1: Set up variables

Let the vertical height of the printed area be \( y \) and the horizontal width of the printed area be \( x \). We know that \( xy = 50\), so \( y=\frac{50}{x}\).
The vertical height of the poster \( h=y + 4\) (2 - inch margin at top and bottom) and the horizontal width of the poster \( w=x + 2\) (1 - inch margin on each side).
The area of the poster \( A=(x + 2)(y + 4)\). Substitute \( y=\frac{50}{x}\) into the area formula: \(A=(x + 2)(\frac{50}{x}+4)=\frac{100}{x}+4x + 58\).

Step2: Find the derivative of the area function

Differentiate \(A(x)\) with respect to \(x\). Using the power rule \((u/v)^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\) and \((ax^{n})^\prime=anx^{n - 1}\), we have \(A^\prime(x)=-\frac{100}{x^{2}}+4\).

Step3: Find the critical points

Set \(A^\prime(x) = 0\). Then \(-\frac{100}{x^{2}}+4 = 0\).

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Step4: Find the width of the poster

Since \(w=x + 2\) and \(x = 5\), then \(w=5 + 2=7\).

Answer:

\(w = 7\) in.