Final answer:
After calculating the area of the border using a 2.5-inch width, it was determined that it does not result in the required 48 square inch area. Thus, a 2.5-inch border is not reasonable for the menu design. The correct answer to the question is 'No'.
Step-by-step explanation:
To determine if a 2.5-inch wide border around the menu is reasonable, we need to calculate the area of the border and see if it equals 48 square inches. Let's use the information provided: the food listings area is a 13-inch by 9-inch rectangle.
If the width of the border is x inches, then the overall dimensions of the menu including the border will be (13 + 2x) inches by (9 + 2x) inches, because the border will be added to all four sides of the menu.
The area of the entire menu, including food listings and border, is calculated by multiplying the length by the width: ({13 + 2x})({9 + 2x}) square inches. To find only the area of the border, we subtract the area of the food listings from this total area:
- Total area of menu including border = (13 + 2x)(9 + 2x)
- Area of food listings = 13 × 9 = 117 square inches
- Area of border = Total area - Area of food listings
To find x when the border area is 48 square inches:
- Area of border = (13 + 2x)(9 + 2x) - 117 = 48
- Expand the equation: (13 + 2x)(9 + 2x) = 165
- Simplify: 117 + 18x + 26x + 4x² = 165
- Combine like terms: 4x² + 44x + 117 = 165
- Subtract 165 from both sides: 4x² + 44x - 48 = 0
This is a quadratic equation, which can be solved for x to find the width of the border. When x = 2.5, let's check:
- 4(2.5)² + 44(2.5) - 48
- 4(6.25) + 110 - 48
- 25 + 110 - 48
- 135 - 48 = 87
Since the equation does not equal zero, a 2.5-inch border width does not give us the required 48 square inch area for the border. Therefore, it is not reasonable for the border to be 2.5 inches wide. The answer is No.