1. **Problem statement:** Find the total surface area and volume of the three composite solids described in Figure 6.53. --- ### a) Cylinder with a truncated cone on top 2. **Given:** - Cylinder: radius $r=3$ cm, height $h=8$ cm - Truncated cone: bottom radius $R=3$ cm, top radius $r=2$ cm, height $h=5$ cm 3. **Formulas:** - Cylinder volume: $V_{cyl} = \pi r^2 h$ - Cylinder surface area (side only): $A_{cyl} = 2 \pi r h$ - Truncated cone volume: $V_{cone} = \frac{1}{3} \pi h (R^2 + Rr + r^2)$ - Truncated cone lateral surface area: $A_{cone} = \pi (R + r) s$ where $s = \sqrt{(R-r)^2 + h^2}$ is the slant height - Total surface area excludes the base of the truncated cone (which sits on the cylinder) and the bottom base of the cylinder is included. 4. **Calculations:** - Cylinder volume: $V_{cyl} = \pi \times 3^2 \times 8 = 72\pi$ cm$^3$ - Cylinder lateral area: $A_{cyl} = 2 \pi \times 3 \times 8 = 48\pi$ cm$^2$ - Cylinder base area: $A_{base} = \pi \times 3^2 = 9\pi$ cm$^2$ - Slant height of truncated cone: $s = \sqrt{(3-2)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}$ cm - Truncated cone volume: $V_{cone} = \frac{1}{3} \pi \times 5 (3^2 + 3 \times 2 + 2^2) = \frac{5\pi}{3} (9 + 6 + 4) = \frac{5\pi}{3} \times 19 = \frac{95\pi}{3}$ cm$^3$ - Truncated cone lateral area: $A_{cone} = \pi (3 + 2) \sqrt{26} = 5\pi \sqrt{26}$ cm$^2$ 5. **Total volume:** $$V = V_{cyl} + V_{cone} = 72\pi + \frac{95\pi}{3} = \frac{216\pi + 95\pi}{3} = \frac{311\pi}{3} \approx 325.9 \text{ cm}^3$$ 6. **Total surface area:** - Sum lateral areas plus bottom base of cylinder (top base of cylinder covered by truncated cone, top base of cone is open) $$A = A_{cyl} + A_{cone} + A_{base} = 48\pi + 5\pi \sqrt{26} + 9\pi = (57\pi + 5\pi \sqrt{26}) \approx 179.1 \text{ cm}^2$$ --- ### b) Rectangular prism with triangular prism on top 7. **Given:** - Rectangular prism dimensions: $5$ cm by $3$ cm by $6$ cm - Triangular prism on top with base lengths $2$ cm and $2.5$ cm, slant height $2.5$ cm 8. **Formulas:** - Rectangular prism volume: $V_{rect} = lwh$ - Rectangular prism surface area: $A_{rect} = 2(lw + lh + wh)$ - Triangular prism volume: $V_{tri} = \text{area of triangle} \times \text{length}$ - Triangular prism surface area: sum of two triangle areas plus three rectangular sides 9. **Calculations:** - Rectangular prism volume: $V_{rect} = 5 \times 3 \times 6 = 90$ cm$^3$ - Rectangular prism surface area: $A_{rect} = 2(5\times3 + 5\times6 + 3\times6) = 2(15 + 30 + 18) = 2 \times 63 = 126$ cm$^2$ 10. **Triangular prism base triangle:** - Base sides: $2$ cm and $2.5$ cm, slant height $2.5$ cm (assumed height of triangle) - Area of triangle: $A_{tri} = \frac{1}{2} \times 2 \times 2.5 = 2.5$ cm$^2$ 11. **Triangular prism volume:** - Length (depth) same as rectangular prism width $= 3$ cm - $V_{tri} = 2.5 \times 3 = 7.5$ cm$^3$ 12. **Triangular prism surface area:** - Two triangle faces: $2 \times 2.5 = 5$ cm$^2$ - Three rectangular faces: - $2 \times 3 = 6$ cm$^2$ - $2.5 \times 3 = 7.5$ cm$^2$ - $2.5 \times 3 = 7.5$ cm$^2$ - Total triangular prism area: $5 + 6 + 7.5 + 7.5 = 26$ cm$^2$ 13. **Total volume:** $$V = V_{rect} + V_{tri} = 90 + 7.5 = 97.5 \text{ cm}^3$$ 14. **Total surface area:** - Remove the rectangular prism top area covered by the triangular prism: top area of rectangular prism is $5 \times 3 = 15$ cm$^2$ - Total surface area: $$A = A_{rect} - 15 + 26 = 126 - 15 + 26 = 137 \text{ cm}^2$$ --- ### c) Rectangular prism with triangular pyramid on top 15. **Given:** - Rectangular prism dimensions: $12$ m by $7$ m by $4$ m - Triangular pyramid on top with height $5$ m and base side length $5$ m (equilateral triangle assumed) 16. **Formulas:** - Rectangular prism volume: $V_{rect} = lwh$ - Rectangular prism surface area: $A_{rect} = 2(lw + lh + wh)$ - Triangular pyramid volume: $V_{pyr} = \frac{1}{3} \times \text{area of base} \times \text{height}$ - Triangular pyramid surface area: base area plus 3 triangular faces 17. **Calculations:** - Rectangular prism volume: $V_{rect} = 12 \times 7 \times 4 = 336$ m$^3$ - Rectangular prism surface area: $A_{rect} = 2(12\times7 + 12\times4 + 7\times4) = 2(84 + 48 + 28) = 2 \times 160 = 320$ m$^2$ 18. **Triangular pyramid base area:** - Equilateral triangle side $a=5$ m - Area: $A_{base} = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \times 25 = \frac{25\sqrt{3}}{4} \approx 10.83$ m$^2$ 19. **Triangular pyramid volume:** $$V_{pyr} = \frac{1}{3} \times 10.83 \times 5 = \frac{1}{3} \times 54.15 = 18.05 \text{ m}^3$$ 20. **Triangular pyramid surface area:** - Base area: $10.83$ m$^2$ - Each triangular face area: height of face $h_f = \sqrt{5^2 - (2.5)^2} = \sqrt{25 - 6.25} = \sqrt{18.75} \approx 4.33$ m - Area of one face: $\frac{1}{2} \times 5 \times 4.33 = 10.83$ m$^2$ - Three faces total: $3 \times 10.83 = 32.49$ m$^2$ - Total pyramid surface area: $10.83 + 32.49 = 43.32$ m$^2$ 21. **Total surface area:** - Remove the rectangular prism top area covered by the pyramid base: $12 \times 7 = 84$ m$^2$ - Total surface area: $$A = A_{rect} - 84 + 43.32 = 320 - 84 + 43.32 = 279.32 \text{ m}^2$$ 22. **Total volume:** $$V = V_{rect} + V_{pyr} = 336 + 18.05 = 354.05 \text{ m}^3$$ --- **Final answers:** - a) Surface area $\approx 179.1$ cm$^2$, Volume $\approx 325.9$ cm$^3$ - b) Surface area $= 137$ cm$^2$, Volume $= 97.5$ cm$^3$ - c) Surface area $\approx 279.32$ m$^2$, Volume $\approx 354.05$ m$^3$