1. **Problem Statement:** A solid cylinder has radius 18 cm and height 15 cm. A conical hole of radius $r$ cm and depth 12 cm is drilled on one end. The volume of the material removed (the cone) is $\frac{9}{10}$ of the volume of the cylinder. Find: (i) The surface area of the hole. (ii) The radius of a spherical ball made from the removed material. 2. **Formulas and Rules:** - Volume of cylinder: $$V_{cyl} = \pi r_{cyl}^2 h_{cyl}$$ - Volume of cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Surface area of cone (lateral + base): $$A = \pi r l + \pi r^2$$ where $$l = \sqrt{r^2 + h^2}$$ is the slant height. - Volume of sphere: $$V_{sphere} = \frac{4}{3} \pi R^3$$ where $R$ is the radius. 3. **Step (i): Find radius $r$ of the conical hole** - Cylinder volume: $$V_{cyl} = \pi \times 18^2 \times 15 = \pi \times 324 \times 15 = 4860\pi$$ - Volume of cone is $\frac{9}{10}$ of cylinder volume: $$V_{cone} = \frac{9}{10} \times 4860\pi = 4374\pi$$ - Volume of cone formula: $$V_{cone} = \frac{1}{3} \pi r^2 \times 12 = 4\pi r^2$$ - Equate and solve for $r$: $$4\pi r^2 = 4374\pi \implies 4 r^2 = 4374 \implies r^2 = \frac{4374}{4} = 1093.5$$ - Radius: $$r = \sqrt{1093.5} \approx 33.07 \text{ cm}$$ 4. **Step (ii): Surface area of the hole (cone)** - Slant height: $$l = \sqrt{r^2 + h^2} = \sqrt{33.07^2 + 12^2} = \sqrt{1093.5 + 144} = \sqrt{1237.5} \approx 35.18 \text{ cm}$$ - Surface area: $$A = \pi r l + \pi r^2 = \pi \times 33.07 \times 35.18 + \pi \times 1093.5$$ $$= \pi (1163.3 + 1093.5) = \pi \times 2256.8 \approx 7088.5 \text{ cm}^2$$ 5. **Step (iii): Radius of spherical ball from removed material** - Volume of material removed = volume of cone = $4374\pi$ - Volume of sphere: $$\frac{4}{3} \pi R^3 = 4374\pi \implies \frac{4}{3} R^3 = 4374 \implies R^3 = \frac{4374 \times 3}{4} = 3280.5$$ - Radius: $$R = \sqrt[3]{3280.5} \approx 14.82 \text{ cm}$$ 6. **Problem (b): Volume of water in rectangular tank** - Dimensions: length = 4.5 m, width = 4 m, height = 2.5 m - Volume: $$V = l \times w \times h = 4.5 \times 4 \times 2.5 = 45 \text{ m}^3$$ - 1 cubic meter = 1 kilolitre, so volume in kilolitres = 45 kL **Final answers:** (i) Surface area of hole: approximately 7088.5 cm² (ii) Radius of spherical ball: approximately 14.82 cm (b) Volume of water in tank: 45 kilolitres