1. **Identify and calculate surface area and volume of solids:** **(1) Cone:** Given height $h=30$ units and radius $r=13$ units. - Surface area $= \pi r (r + l)$, where $l=\sqrt{h^2 + r^2}=\sqrt{30^2 + 13^2}=\sqrt{900 + 169}=\sqrt{1069}\approx 32.7$. - Surface area $= \pi \times 13 \times (13 + 32.7) \approx 3.1416 \times 13 \times 45.7 \approx 1866.2$ square units. - Volume $= \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 13^2 \times 30 = \frac{1}{3} \pi \times 169 \times 30 = \frac{1}{3} \times 3.1416 \times 5070 \approx 5316.1$ cubic units. **(2) Rectangular prism:** Length $8$ km, width $6$ km, height $4$ km. - Surface area $= 2(lw + lh + wh) = 2(8 \times 6 + 8 \times 4 + 6 \times 4) = 2(48 + 32 + 24) = 2 \times 104 = 208$ km². - Volume $= l \times w \times h = 8 \times 6 \times 4 = 192$ km³. **(3) Pentagonal pyramid:** Base side length $6$ in, height $12.2$ in, apothem $5.2$ in. - Base perimeter $P = 5 \times 6 = 30$ in. - Base area $= \frac{1}{2} P \times $ apothem $= \frac{1}{2} \times 30 \times 5.2 = 78$ in². - Volume $= \frac{1}{3} \times$ base area $\times$ height $= \frac{1}{3} \times 78 \times 12.2 = 317.2$ in³. - Surface area $= $ base area + lateral area. - Lateral area $= \frac{1}{2} \times P \times$ slant height. Slant height $l = \sqrt{12.2^2 + 5.2^2} = \sqrt{148.84 + 27.04} = \sqrt{175.88} \approx 13.26$ in. - Lateral area $= \frac{1}{2} \times 30 \times 13.26 = 198.9$ in². - Surface area $= 78 + 198.9 = 276.9$ in². 4. **Cylinder:** height $8$ km, base diameter $14$ km implies radius $r=7$ km. - Surface area $= 2 \pi r (r + h) = 2 \times 3.1416 \times 7 \times (7 + 8) = 14 \times 3.1416 \times 15 = 659.7$ km². - Volume $= \pi r^2 h = 3.1416 \times 7^2 \times 8 = 3.1416 \times 49 \times 8 = 1231.5$ km³. 5. **Sphere:** radius $8$ in. - Surface area $= 4 \pi r^2 = 4 \times 3.1416 \times 64 = 804.25$ in². - Volume $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.1416 \times 512 = 2144.66$ in³. 6. **Pentagonal prism:** side length $12$ yd, height $8$ yd, apothem $8.3$ yd. - Base perimeter $P = 5 \times 12 = 60$ yd. - Base area $= \frac{1}{2} P \times $ apothem $= \frac{1}{2} \times 60 \times 8.3 = 249$ yd². - Surface area $= 2 \times$ base area $+ P \times$ height $= 2 \times 249 + 60 \times 8 = 498 + 480 = 978$ yd². - Volume $= $ base area $\times$ height $= 249 \times 8 = 1992$ yd³. 2. **Solve the following problems:** 1. Rectangular prism $6$ cm by $4$ cm by $4$ cm, cubes with edge $2$ cm. - Volume prism $= 6 \times 4 \times 4 = 96$ cm³. - Volume cube $= 2^3 = 8$ cm³. - Number of cubes $= 96 \div 8 = 12$ cubes. 2. Aquarium $2$ m by $3$ m by $4$ m. - Volume $= 2 \times 3 \times 4 = 24$ m³. - $1$ m³ $= 1000$ liters, so volume in liters $= 24 \times 1000 = 24000$ liters. 3. Balloon diameter $18$ cm, radius $9$ cm. - Volume $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.1416 \times 9^3 = \frac{4}{3} \times 3.1416 \times 729 = 3053.6$ cm³. 4. Cylinder and cone same diameter and height, cone volume $36.5$ cm³. - Volume cone $= \frac{1}{3} \pi r^2 h = 36.5$ cm³. - Volume cylinder $= \pi r^2 h = 3 \times 36.5 = 109.5$ cm³. 5. Sphere radius $4$ cm inside cylinder height $8$ cm with same radius. - Volume cylinder $= \pi r^2 h = 3.1416 \times 4^2 \times 8 = 402.12$ cm³. - Volume sphere $= \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.1416 \times 64 = 268.08$ cm³. - Volume inside cylinder but outside sphere $= 402.12 - 268.08 = 134.04$ cm³. 6. Cylindrical can diameter $10$ cm, height $20$ cm. Sand height $18$ cm. - Radius $r=5$ cm. - Volume sand $= \pi r^2 h = 3.1416 \times 5^2 \times 18 = 1413.72$ cm³. - Rectangular container $10 \times 30 \times 20 = 6000$ cm³. - Empty space volume $= 6000 - 1413.72 = 4586.28$ cm³. **Final answers:** 1. Cone surface area $\approx 1866.2$, volume $\approx 5316.1$. 2. Rectangular prism surface area $= 208$, volume $= 192$. 3. Pentagonal pyramid surface area $\approx 276.9$, volume $\approx 317.2$. 4. Cylinder surface area $\approx 659.7$, volume $\approx 1231.5$. 5. Sphere surface area $\approx 804.25$, volume $\approx 2144.66$. 6. Pentagonal prism surface area $= 978$, volume $= 1992$. Problem B answers: 1. $12$ cubes. 2. $24000$ liters. 3. Volume of balloon $\approx 3053.6$ cm³. 4. Volume of cylinder $= 109.5$ cm³. 5. Volume between cylinder and sphere $\approx 134.04$ cm³. 6. Empty space volume $\approx 4586.28$ cm³.