Final answer:
The width of the walkway around the rectangular garden is 3 feet. The area of the garden and walkway together is 320 square feet, and after setting up and solving a quadratic equation, we determine the walkway's width.
Step-by-step explanation:
To find the width of the walkway around the rectangular garden, we need to first calculate the area of just the garden, which is the length multiplied by the width: 10 feet (width) × 14 feet (length) = 140 square feet.
Since the area of the garden plus the walkway is 320 square feet, we subtract the area of the garden from this to find the area that the walkway occupies: 320 square feet − 140 square feet = 180 square feet.
The walkway surrounds the entire garden, so if we let 'w' denote the width of the walkway, the overall dimensions of the garden plus the walkway will be (10 + 2w) feet in width and (14 + 2w) feet in length. So the area is (10 + 2w) × (14 + 2w) = 320 square feet.
Expanding this, we get:
10 × 14 + 2w × 10 + 2w × 14 + 4w^2 = 320
140 + 20w + 28w + 4w^2 = 320
4w^2 + 48w + 140 = 320
4w^2 + 48w − 180 = 0
Dividing by 4, we simplify the quadratic equation to:
w^2 + 12w − 45 = 0
Solving the quadratic equation, we find that w = 3 (rejecting the negative solution for w).
Therefore, the width of the walkway is 3 feet.