1. **Problem I:** Calculate the cost of painting the inner side of a tent shaped as a cylinder (height 3m, radius 14m) surmounted by a cone (height 13.5m, radius 14m) at the rate of 2 per m². 2. Calculate the surface area of the cylindrical part: $$A_{cyl} = 2\pi r h = 2 \times 3.14 \times 14 \times 3 = 263.76\,m^2$$ 3. Calculate the slant height of the cone: $$l = \sqrt{r^2 + h^2} = \sqrt{14^2 + 10.5^2} = \sqrt{196 + 110.25} = \sqrt{306.25} = 17.5\,m$$ 4. Calculate the lateral surface area of the cone: $$A_{cone} = \pi r l = 3.14 \times 14 \times 17.5 = 769.7\,m^2$$ 5. Total inner surface area to paint: $$A_{total} = A_{cyl} + A_{cone} = 263.76 + 769.7 = 1033.46\,m^2$$ 6. Cost of painting: $$Cost = 1033.46 \times 2 = 2066.92$$ --- 7. **Problem II:** From a solid cylinder (height 10cm, radius 6cm), a cone of same height and base is removed. Find the whole surface area. 8. Surface area of cylinder (excluding top base): $$A_{cyl} = 2\pi r h + \pi r^2 = 2 \times 3.14 \times 6 \times 10 + 3.14 \times 6^2 = 376.8 + 113.04 = 489.84\,cm^2$$ 9. Lateral surface area of cone: $$l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \approx 11.66\,cm$$ $$A_{cone} = \pi r l = 3.14 \times 6 \times 11.66 = 219.7\,cm^2$$ 10. Whole surface area after removing cone (cylinder base + curved surface of cone + curved surface of cylinder): $$A_{total} = \pi r^2 + A_{cone} + 2\pi r h = 113.04 + 219.7 + 376.8 = 709.54\,cm^2$$ --- 11. **Problem III:** A solid composed of a cylinder with hemispherical ends. Total length 108cm, diameter of hemispheres 36cm. Find cost of polishing at 0.07 per sq.cm. 12. Radius of hemispheres and cylinder: $$r = \frac{36}{2} = 18\,cm$$ 13. Length of cylinder: $$h = 108 - 2r = 108 - 36 = 72\,cm$$ 14. Surface area of cylinder: $$A_{cyl} = 2\pi r h = 2 \times 3.14 \times 18 \times 72 = 8145.9\,cm^2$$ 15. Surface area of two hemispheres (which make a sphere): $$A_{sphere} = 4\pi r^2 = 4 \times 3.14 \times 18^2 = 4071.5\,cm^2$$ 16. Total surface area: $$A_{total} = A_{cyl} + A_{sphere} = 8145.9 + 4071.5 = 12217.4\,cm^2$$ 17. Cost of polishing: $$Cost = 12217.4 \times 0.07 = 855.22$$ --- 18. **Problem IV:** Block made of a cone surmounted by a hemisphere. Cone height 24cm, radius 10cm, hemisphere diameter 10cm. 19. Radius of hemisphere: $$r = \frac{10}{2} = 5\,cm$$ 20. Surface area of hemisphere: $$A_{hem} = 2\pi r^2 = 2 \times 3.14 \times 5^2 = 157\,cm^2$$ 21. Slant height of cone: $$l = \sqrt{r^2 + h^2} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26\,cm$$ 22. Lateral surface area of cone: $$A_{cone} = \pi r l = 3.14 \times 10 \times 26 = 816.4\,cm^2$$ 23. Total surface area (excluding base of cone covered by hemisphere): $$A_{total} = A_{hem} + A_{cone} = 157 + 816.4 = 973.4\,cm^2$$ --- 24. **Problem V:** Block made of cube (edge 6cm) and hemisphere (diameter 4.9cm) on top. 25. Surface area of cube (all faces): $$A_{cube} = 6a^2 = 6 \times 6^2 = 216\,cm^2$$ 26. Radius of hemisphere: $$r = \frac{4.9}{2} = 2.45\,cm$$ 27. Surface area of hemisphere (curved surface only): $$A_{hem} = 2\pi r^2 = 2 \times 3.14 \times 2.45^2 = 37.7\,cm^2$$ 28. Base of hemisphere covers one face of cube, so subtract one face area: $$A_{total} = A_{cube} - a^2 + A_{hem} = 216 - 36 + 37.7 = 217.7\,cm^2$$ **Final answers:** I. Cost = 2066.92 II. Whole surface area = 709.54 cm² III. Cost = 855.22 IV. Total surface area = 973.4 cm² V. Total surface area = 217.7 cm²