1. **Problem 1: Total Surface Area of the Bird-bath** The bird-bath consists of a cylinder with a hemispherical depression at one end. 2. **Given:** - Cylinder height $h = 1.45$ m - Radius $r = 30$ cm = 0.30 m 3. **Formulas:** - Curved surface area of cylinder: $2\pi r h$ - Surface area of hemisphere: $2\pi r^2$ - Total surface area = curved surface area of cylinder + curved surface area of hemisphere + base area of cylinder (since the hemisphere is a depression, the top circular area is replaced by the hemisphere's inner surface) 4. **Calculate each part:** - Curved surface area of cylinder: $2\pi (0.30)(1.45) = 2\pi \times 0.435 = 2.732$ m$^2$ (approx) - Surface area of hemisphere: $2\pi (0.30)^2 = 2\pi \times 0.09 = 0.565$ m$^2$ (approx) - Base area of cylinder: $\pi r^2 = \pi (0.30)^2 = 0.283$ m$^2$ (approx) 5. **Total surface area:** $$\text{Total} = 2.732 + 0.565 + 0.283 = 3.58 \text{ m}^2$$ --- 6. **Problem 2: Volume and Total Surface Area of the Square Pyramid** 7. **Given:** - Base edge $a = 5$ cm - Perpendicular height $h = 12$ cm 8. **Formulas:** - Volume of pyramid: $V = \frac{1}{3} a^2 h$ - Slant height $l = \sqrt{\left(\frac{a}{2}\right)^2 + h^2}$ - Surface area: $A = a^2 + 2 a l$ 9. **Calculate slant height:** $$l = \sqrt{\left(\frac{5}{2}\right)^2 + 12^2} = \sqrt{2.5^2 + 144} = \sqrt{6.25 + 144} = \sqrt{150.25} = 12.26 \text{ cm (approx)}$$ 10. **Calculate volume:** $$V = \frac{1}{3} \times 5^2 \times 12 = \frac{1}{3} \times 25 \times 12 = 100 \text{ cm}^3$$ 11. **Calculate surface area:** $$A = 5^2 + 2 \times 5 \times 12.26 = 25 + 122.6 = 147.6 \text{ cm}^2$$ **Final answers:** - Bird-bath total surface area: $3.58$ m$^2$ - Square pyramid volume: $100$ cm$^3$ - Square pyramid total surface area: $147.6$ cm$^2$