Question

Johnny is designing a rectangular poster to contain 10 in^2 of printing with a 2​-in margin at the top and bottom and a 5​-in margin at each side. What overall dimensions will minimize the amount of paper​ used?

Answer 1

To minimize the amount of paper used, we need to find the dimensions that minimize the total area of the poster.

Let's assume the length of the poster is x inches. Since there are 5-inch margins on each side, the printable width of the poster will be x - 10 inches.

The width of the poster can be found by subtracting the top and bottom margins from the length: x - (2 + 2) = x - 4 inches.

The area of the poster can be calculated by multiplying the length and width: Area = (x - 4)(x - 10).

To minimize the area, we need to find the value of x that minimizes the area function. Taking the derivative of the area function with respect to x and setting it equal to zero, we can find the critical points.

d(Area)/dx = 2x - 14x + 40 = -12x + 40

Setting -12x + 40 = 0 and solving for x:

-12x = -40

x = 40/12

x ≈ 3.33

Since the length of the poster cannot be negative, we discard the negative solution.

Therefore, the length of the poster that minimizes the amount of paper used is approximately 3.33 inches.

To find the corresponding width, we substitute this value back into the width equation: x - 4 ≈ 3.33 - 4 = -0.67 inches.

However, since the width cannot be negative, we discard this solution as well.

In conclusion, the dimensions that minimize the amount of paper used are approximately 3.33 inches for the length and 0.67 inches for the width (taking into account that negative dimensions are not valid in this context).

Answer 2

To minimize the amount of paper used for the rectangular poster, we can start by calculating the area of the printed portion.

The printed portion has an area of 10 in^2. Since there are 2-in margins at the top and bottom, the height of the printed portion will be 10 in - (2 in + 2 in) = 6 in.

Since there are 5-in margins on each side, the width of the printed portion will be the total width of the poster minus the combined width of the margins, which is 10 in - (5 in + 5 in) = 0 in.

However, we cannot have a width of 0 in, so we need to adjust the dimensions of the poster to maintain the same ratio while minimizing the paper used.

Let's assume the overall dimensions of the poster are w inches (width) and h inches (height). The width of the printed portion will be w - (5 in + 5 in) = w - 10 in.

Since the ratio of the width to height of the printed portion must remain the same (6 in : (w - 10 in)), we can set up the following equation:

6 in / (w - 10 in) = 10 in^2 / (w * h)

Now we can solve this equation to find the optimal overall dimensions of the poster. However, without knowing any specific values for w or h, it is not possible to provide a specific answer.