Final answer:
The width of the border around the bulletin board is 3.2 feet, calculated by solving the quadratic equation derived from the total area of the bulletin board with border and the area of the bulletin board itself.
Step-by-step explanation:
To calculate the width of the border that will be placed around the bulletin board, we need to find the difference between the total area (board plus border) and the area of the board itself. First, we figure out the area of the bulletin board by multiplying the length by the width: Area of bulletin board = 10 ft x 4 ft = 40 ft².
Next, we subtract the area of the bulletin board from the total area to find the area of the border: Area of border = 156 ft² - 40 ft² = 116 ft².
To find the width of the border, we need to see this as a frame with equal widths around the board. The board's length with the border becomes 10 ft + 2b (where b is the width of the border) and the width becomes 4 ft + 2b. The total area is then expressed by the equation:
(10 ft + 2b) x (4 ft + 2b) = 156 ft².
Expanding this and creating a quadratic equation we have:
40 ft² + 20b + 8b + 4b² = 156 ft²
Combine like terms:
4b² + 28b + 40 ft² - 156 ft² = 0
Simplifying further:
4b² + 28b - 116 = 0
Divide the entire equation by 4 to simplify:
b² + 7b - 29 = 0
Solving this quadratic equation, we find that b (the width of the border) is 3.2 feet (Only the positive root is a feasible physical solution).