Final answer:
To find the width of the path around the rectangular field, subtract the area of the field from the total area of the field with the path. The width of the path is 5m.
Step-by-step explanation:
To find the width of the path around the rectangular field, we need to subtract the area of the field from the total area of the field with the path. The total area of the field with the path is the sum of the area of the field and the area of the path.
The area of the rectangular field is found by multiplying its length by its width. In this case, the length is 16m and the width is 10m, so the area of the field is 16m * 10m = 160m².
We are given that the total area of the field with the path is 120m² more than the area of the field. So we can set up the equation: Total area = 160m² + 120m²
Simplifying the equation gives us: Total area = 280m²
Now we need to find the width of the path. Let's assume that the width of the path is 'x' meters. Since the path is uniform in width all around the field, we can subtract 2 times the width of the path from both the length and the width of the field to get the new dimensions. The length of the field becomes 16m - 2x and the width becomes 10m - 2x.
The area of the field with the path is equal to the new length times the new width. So we can set up the equation: Total area = (16m - 2x)(10m - 2x)
We already know that the total area is 280m², so we can substitute that into the equation: 280m² = (16m - 2x)(10m - 2x)
Expanding the equation gives us: 280m² = 160m² - 44xm + 4x²
Combining like terms gives us: 0 = 4x² - 44xm + 120m²
To solve this quadratic equation, we can use factoring or the quadratic formula. Factoring is simpler in this case, so let's factor the equation: 0 = (2x - 10m)(2x - 12m)
Setting each factor equal to zero gives us two possible solutions: 2x - 10m = 0 or 2x - 12m = 0
Solving each equation gives us x = 5m or x = 6m. Since the width of the path cannot be negative, the width of the path is 5m.