1. **Stating the problem:** Convert between different units of volume and measure, compare values, and arrange volumes in increasing order. --- ### Example 3: Volume unit conversions 2. **Formula and rules:** - $1\text{ km}^3 = 1000^3 = 10^9\text{ m}^3$ - $1\text{ hm}^3 = 100^3 = 10^6\text{ m}^3$ - $1\text{ dm}^3 = 1\text{ L} = 1000\text{ cm}^3$ - $1\text{ cm}^3 = 1000\text{ mm}^3$ 3. **Conversions:** **a) Convert 3 km³ to hm³:** $$3\text{ km}^3 = 3 \times 1000\text{ hm}^3 = 3000\text{ hm}^3$$ **b) Convert 0.45 cm³ to mm³:** $$0.45\text{ cm}^3 = 0.45 \times 1000\text{ mm}^3 = 450\text{ mm}^3$$ **c) Convert 6 hm³ to km³:** $$6\text{ hm}^3 = 6 \times \frac{1}{1000}\text{ km}^3 = 6 \times \frac{1}{\cancel{1000}}\cancel{\text{ km}^3} = \frac{6}{1000} = 0.006\text{ km}^3$$ **d) Convert 8 cm³ to dm³:** $$8\text{ cm}^3 = 8 \times \frac{1}{1000}\text{ dm}^3 = \frac{8}{1000} = 0.008\text{ dm}^3$$ --- ### Example 4: Volume to milliliters and liters 4. **Formula and rules:** - $1\text{ cm}^3 = 1\text{ mL}$ - $1\text{ mm}^3 = 0.001\text{ mL}$ - $1\text{ dm}^3 = 1\text{ L}$ - $1\text{ m}^3 = 1000\text{ L}$ 5. **Conversions:** **a) 23 cm³ to mL:** $$23\text{ cm}^3 = 23\text{ mL}$$ **b) 4792 mm³ to mL:** $$4792\text{ mm}^3 = 4792 \times 0.001\text{ mL} = 4.792\text{ mL}$$ **c) 178 dm³ to L:** $$178\text{ dm}^3 = 178\text{ L}$$ **d) 267 m³ to L:** $$267\text{ m}^3 = 267 \times 1000\text{ L} = 267000\text{ L}$$ --- ### Example 5: Comparing units 6. **Convert to same units and compare:** **a) 2 m and 20 dm:** $$20\text{ dm} = 20 \times 0.1\text{ m} = 2\text{ m}$$ So, $2\text{ m} = 20\text{ dm}$ **b) 14 cm and 1 dam:** $$1\text{ dam} = 10\text{ m} = 1000\text{ cm}$$ So, $14\text{ cm} < 1000\text{ cm}$ **c) 215 m² and 12 dam²:** $$12\text{ dam}^2 = 12 \times (10)^2 = 12 \times 100 = 1200\text{ m}^2$$ So, $215 < 1200$ **d) 18 km² and 450000 m²:** $$18\text{ km}^2 = 18 \times (1000)^2 = 18 \times 1,000,000 = 18,000,000\text{ m}^2$$ So, $18,000,000 > 450,000$ --- ### Example 6: Arrange volumes in increasing order 7. **Convert all to dm³ or L (1 dm³ = 1 L):** **a)** - $347\text{ dm}^3 = 347$ L - $0.45\text{ m}^3 = 0.45 \times 1000 = 450$ L - $58\text{ L} = 58$ L - $56,345\text{ cm}^3 = 56,345 \times 0.001 = 56.345$ L Order: $56.345 < 58 < 347 < 450$ **b)** - $12\text{ cm}^3 = 0.012$ L - $21,670\text{ mm}^3 = 21,670 \times 0.001 = 21.67$ mL = 0.02167 L - $51\text{ mL} = 0.051$ L - $0.03\text{ L} = 0.03$ L Order: $0.012 < 0.02167 < 0.03 < 0.051$ --- **Final answers:** - Example 3: a) 3000 hm³, b) 450 mm³, c) 0.006 km³, d) 0.008 dm³ - Example 4: a) 23 mL, b) 4.792 mL, c) 178 L, d) 267000 L - Example 5: a) =, b) <, c) <, d) > - Example 6: a) 56,345 cm³ < 58 L < 347 dm³ < 0.45 m³ b) 12 cm³ < 21,670 mm³ < 0.03 L < 51 mL