you are designing a rectangular poster to contain 50 in.² of printing with a 2 - in. margin at the top and…

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). [ \begin{align*} 4&=\frac{100}{x^{2}}\ x^{2}& = 25\ x&=5\quad(x>0) \end{align*} ]

Step4: Find the width of the poster

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

Answer:

(w = 7) in.