Answer :
Final answer:
The width of the path around the rectangular field is determined to be 3 meters by setting up a quadratic equation based on the areas and solving for the uniform width w. Thus, the width of the path is 3 meters.
Explanation:
To find the width of the path around the rectangular field, let's denote the uniform width of the path as w meters. The dimensions of the field including the path would then be (16 + 2w) meters in length and (10 + 2w) meters in width. The area of the path can be calculated by subtracting the area of the field from the area of the larger rectangle that includes the field and the path. The area of the field is 16m × 10m = 160m2.
The area of the larger rectangle is (16 + 2w)m × (10 + 2w)m. Since the area of the path is given to be 120m2, we set up the equation:
(16 + 2w) × (10 + 2w) - 160m2 = 120m2
After expanding and simplifying, we get a quadratic equation:
4w2 + 52w - 120 = 0
By solving the quadratic equation for w, we can find the width of the path. Factoring, or using the quadratic formula yields:
w = 3 meters or a negative value which is not applicable for this scenario. Thus, the width of the path is 3 meters.
