1. **Problem Statement:** Find the area of the grassy border around a circular fountain with radius 5 m, where the grass border extends to 8 m from the center. 2. **Formula:** The surface area of a ring (annulus) is given by: $$\text{Area} = \pi (R^2 - r^2)$$ where $R$ is the outer radius and $r$ is the inner radius. 3. **Step-by-step solution:** - Given: $r = 5$ m (fountain radius), $R = 8$ m (outer radius of grass border). - Calculate the area of the outer circle: $$\pi R^2 = \pi \times 8^2 = 64\pi$$ - Calculate the area of the inner circle (fountain): $$\pi r^2 = \pi \times 5^2 = 25\pi$$ - Area of grassy border = outer area - inner area: $$64\pi - 25\pi = 39\pi$$ - Approximate using $\pi \approx 3.14$: $$39 \times 3.14 = 122.46 \text{ m}^2$$ 4. **Grass carpet needed:** - Each carpet roll covers 4 m². - Number of rolls needed = total grassy area / area per roll: $$\frac{122.46}{4} = 30.615$$ - Since rolls cannot be fractional, round up to 31 rolls. **Final answers:** - Area of grassy border: $122.46$ m² - Carpet rolls needed: 31 rolls 5. **Additional problems:** **Playground Track:** - Outer radius $R=14$ m, inner radius $r=7$ m. - Area = $\pi (14^2 - 7^2) = \pi (196 - 49) = 147\pi \approx 461.58$ m². **Table Mat Design:** - Outer radius $R=10$ cm, inner radius $r=6$ cm. - Area = $\pi (10^2 - 6^2) = \pi (100 - 36) = 64\pi \approx 201.06$ cm². **Circular Road:** - Outer radius $R=60$ m, inner radius $r=50$ m. - Area = $\pi (60^2 - 50^2) = \pi (3600 - 2500) = 1100\pi \approx 3454$ m². - Cost = area × cost per m² = $3454 \times 150 = 518100$. **Decorative Mirror Frame:** - Outer radius $R=25$ cm, inner radius $r=20$ cm. - Area = $\pi (25^2 - 20^2) = \pi (625 - 400) = 225\pi \approx 706.86$ cm².