a rectangular peice of land is s20m long and 14m wide. a path of uniform width and 111m^2 area surround's the land frm outside. find width

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We have a rectangular piece of land that is 20 meters long and 14 meters wide. A path of uniform width surrounds this land from the outside. The area of this path is 111 square meters. We need to find the width of the path, which we'll call x. Let's set up our equation step by step. First, we define x as the width of the path. The land area is 20 times 14, which equals 280 square meters. When we add the path, the total length becomes 20 plus 2x, and the total width becomes 14 plus 2x. The total area is the product of these dimensions. Since the path area equals the total area minus the land area, we get the equation: 111 equals the product of 20 plus 2x and 14 plus 2x, minus 280. Now let's expand and simplify our equation. Starting with 111 equals the product of 20 plus 2x and 14 plus 2x, minus 280. When we expand the product, we get 280 plus 40x plus 28x plus 4x squared, minus 280. The 280 terms cancel out, leaving us with 111 equals 4x squared plus 68x. Rearranging this into standard form, we get 4x squared plus 68x minus 111 equals zero. This is a quadratic equation where a equals 4, b equals 68, and c equals negative 111. Now we'll solve the quadratic equation using the quadratic formula. For the equation 4x squared plus 68x minus 111 equals zero, we use the formula x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. With a equals 4, b equals 68, and c equals negative 111, we first calculate the discriminant: 68 squared minus 4 times 4 times negative 111, which equals 4624 plus 1776, giving us 6400. The square root of 6400 is 80. This gives us two solutions: x equals negative 68 plus 80 divided by 8, which equals 1.5, and x equals negative 68 minus 80 divided by 8, which equals negative 18.5. We found two mathematical solutions: 1.5 meters and negative 18.5 meters. However, since the width of a path must be positive, we reject the negative solution. Therefore, the width of the path is 1.5 meters. We can verify this: with a path width of 1.5 meters, the total dimensions become 23 meters by 17 meters, giving a total area of 391 square meters. Subtracting the land area of 280 square meters gives us exactly 111 square meters for the path area, confirming our answer.