Answer :
Final answer:
The width of the path around the rectangular field is 2 meters. This is determined by setting up a quadratic equation to represent the total area of the field with the path and then solving for the width.
Explanation:
To find the width of the path around a rectangular field, we need to calculate the combined area of the path and the field, and then subtract the area of the field itself. The field is 16 meters long and 10 meters wide, while the path has an area of 120 square meters. We let w represent the uniform width of the path.
The total area of the field with the path is the field's area plus the path's area:
- Area of the field: Length × Width = 16m × 10m = 160m2
- Area of the field with path: 160m2 + 120m2 = 280m2
If we add the width of the path to both the length and the width of the field, we have the total length as (16 + 2w) and the total width as (10 + 2w). The equation for the total area is:
(16 + 2w) × (10 + 2w) = 280m2
Expanding this, we get:
160 + 32w + 20w + 4w2 = 280
Combine like terms and subtract 280 from both sides to get a quadratic equation:
4w2 + 52w + 160 = 280
4w2 + 52w - 120 = 0
Now we can solve for w using factorization or the quadratic formula.
Let's factor:
4w2 + 60w - 8w - 120 = 0
4w(w + 15) - 8(w + 15) = 0
(4w - 8)(w + 15) = 0
From this, w can be either 2 or -15. However, width cannot be negative, so:
w = 2 meters
Hence, the uniform width of the path is 2 meters.
