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

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 = □ in.

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 = □ in.

Answer

Answer:

(h = 12) in.

Explanation:

Step1: Define variables

Let the width of the printed area be (x) inches and the height of the printed area be (y) inches. We know that (xy=50), so (y = \frac{50}{x}). The overall width of the poster (W=x + 2) (1 - inch margin on each side) and the overall height of the poster (h=y + 4) (2 - inch margin on top and bottom).

Step2: Express the area of the poster

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).

Step3: 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}), (A^\prime(x)=-\frac{100}{x^{2}}+4).

Step4: Set the derivative equal to zero

Set (A^\prime(x) = 0), so (-\frac{100}{x^{2}}+4 = 0). Then (4x^{2}=100), (x^{2}=25), and (x = 5) (we take (x>0) since (x) represents a length).

Step5: Find the height of the poster

Since (y=\frac{50}{x}), when (x = 5), (y = 10). Then (h=y + 4). Substitute (y = 10) into the formula for (h), we get (h=10 + 2=12) inches.