1. **Problem 1:** Find the volume and total surface area of a right prism with bases 12 cm apart and equilateral triangular base with side 6 cm. 2. The height (distance between bases) is $h=12$ cm. 3. Area of equilateral triangle base with side $a=6$ cm is $$A=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}\times 6^2 = 9\sqrt{3} \text{ cm}^2.$$ 4. Volume $$V = \text{Base area} \times \text{height} = 9\sqrt{3} \times 12 = 108\sqrt{3} \text{ cm}^3.$$ 5. Total surface area includes 2 triangular bases plus 3 rectangular lateral faces. 6. The perimeter of the triangular base is $P=3 \times 6=18$ cm. 7. Lateral surface area $$= P \times h = 18 \times 12 = 216 \text{ cm}^2.$$ 8. Total surface area $$= 2\times 9\sqrt{3} + 216 = 18\sqrt{3} + 216 \text{ cm}^2.$$ --- 9. **Problem 2:** Given right prism with square base, lateral edge 10 cm, lateral area 120 cm$^2$, find volume. 10. Let side of square base be $s$. Height $h=10$ cm (lateral edge). 11. Lateral surface area for prism with square base $$= \text{perimeter of base} \times \text{height} = 4s \times 10 = 40s.$$ 12. Given lateral area $= 120$, so $$40s = 120 \Rightarrow s = 3 \text{ cm}.$$ 13. Volume $$V = \text{area of base} \times height = s^2 \times h = 3^2 \times 10 = 90 \text{ cm}^3.$$ --- 14. **Problem 3:** A slab with two parallel equal bases each a sector, density 10 g/in$^3$, find total mass in kilograms. 15. The description shows a prism with base rectangle 8" by height unknown $h$, topped by a sector with angle $35^\circ48'$ (convert to decimal: $35 + 48/60 = 35.8^\circ$), height of curved sector is $h$, and prism height is 15". 16. Since height is 15", volume equals area of base times height. 17. Area of rectangular base = $8 \times 15 = 120$ in$^2$. 18. The sector area calculation requires radius and angle; assuming radius = 8, sector angle in radians $$\theta=35.8^\circ \times \frac{\pi}{180} = 0.625 \text{ rad}.$$ 19. Area of sector $$A=\frac{\theta}{2\pi} \times \pi r^2 = \frac{\theta}{2} r^2 = \frac{0.625}{2} \times 8^2 = 0.3125 \times 64 = 20 \text{ in}^2.$$ 20. Total base area including sector and rectangle = 120 + 20 = 140 in$^2$ (assuming sector is on top of rectangle). 21. Volume $$V = 140 \times h \text{ in}^3,$$ but $h$ is not provided, assuming $h=1$ for calculation or use given values from problem. 22. Mass $$= \text{density} \times \text{volume} = 10 \times V \text{ grams}.$$ Convert grams to kilograms dividing by 1000. 23. Without explicit height $h$ or further info, exact mass cannot be computed; please clarify $h$ or relevant dimension. **Final answers:** - Volume problem 1: $108\sqrt{3} \approx 187.1$ cm$^3$. - Surface area problem 1: $18\sqrt{3} + 216 \approx 247.2$ cm$^2$. - Volume problem 2: $90$ cm$^3$. - Mass problem 3: insufficient data to compute.