1. **Problem 27: Calculate the surface area of a triangular-based pyramid with each edge measuring 10 cm.** - The pyramid is a regular tetrahedron with 4 equilateral triangle faces. - Each face side length $s=10$ cm. 2. **Calculate the area of one equilateral triangle face.** - Area formula: $$A=\frac{\sqrt{3}}{4}s^2$$ - Substitute $s=10$: $$A=\frac{\sqrt{3}}{4} \times 10^2 = 25\sqrt{3} \approx 43.3 \text{ cm}^2$$ 3. **Calculate total surface area.** - Total area = 4 faces $\times$ area of one face - $$4 \times 25\sqrt{3} = 100\sqrt{3} \approx 173.2 \text{ cm}^2$$ --- 4. **Problem 28: Calculate total cost after duties and VAT on goods worth 3500000.** - Value before duty = 3500000 - Import duty (20%): $$3500000 \times 0.20 = 700000$$ - Excise duty (25%): $$3500000 \times 0.25 = 875000$$ - Total before VAT: $$3500000 + 700000 + 875000 = 5075000$$ - VAT (16%) included, i.e., total cost = 5075000 - The VAT amount is $$5075000 \times \frac{16}{116} \approx 699138$$ - Total cost paid (including VAT) is \textbf{5075000} (since VAT is included in this amount) --- 5. **Problem 29: Calculate percentage error in area from approximating length and width.** - Actual length = 10 m, actual width = 8 m - Approximated length = 9.5 m, approximated width = 7.8 m - Actual area: $$10 \times 8 = 80 \text{ m}^2$$ - Approximated area: $$9.5 \times 7.8 = 74.1 \text{ m}^2$$ - Error: $$|80 - 74.1| = 5.9$$ - Percentage error: $$\frac{5.9}{80} \times 100\% = 7.375\% \approx 7.38\%$$ --- 6. **Problem 30: Calculate density of object with mass 60 kg and volume 0.05 $\text{m}^3$.** - Density formula: $$\rho=\frac{\text{mass}}{\text{volume}}$$ - $$\rho = \frac{60 \text{ kg}}{0.05 \text{ m}^3} = 1200 \text{ kg/m}^3$$ - Convert to g/cm³: $$1 \text{ kg/m}^3 = 0.001 \text{ g/cm}^3$$ - $$1200 \times 0.001 = 1.2 \text{ g/cm}^3$$ --- 7. **Problem 31: Price change in third week in continuous proportion with prices 8, 18, k, 72.** - Continuous proportion means: $$18^2 = 8 \times k$$ and $$k^2 = 18 \times 72$$ - Solve for $k$ using the first relation: $$k = \frac{18^2}{8} = \frac{324}{8} = 40.5$$ - Check with second relation: $$k^2 = 18 \times 72 = 1296$$ $$k = \sqrt{1296} = 36$$ - The two values differ, meaning only one continuous proportion fits either sequence. - For continuous proportion progression, standard definition uses $k^2 = 18 \times 72$; so $k=36$. - Change in price on third week: $$k - 18 = 36 - 18 = 18$$ --- 8. **Problem 32: Find actual land dimensions from map scale 1:100000 and map size 7 cm by 4 cm.** - Scale means 1 cm on map = 100000 cm real - Actual length: $$7 \times 100000 = 700000 \text{ cm} = 7 \text{ km}$$ - Actual width: $$4 \times 100000 = 400000 \text{ cm} = 4 \text{ km}$$ --- 9. **Problem 33: Selling price to gain 20% if sold at 15000 with 10% loss.** - Let cost price = $C$ - Loss 10% means: $$Selling\ Price = 0.9 C = 15000$$ - $$C = \frac{15000}{0.9} = 16666.67$$ - To get 20% profit, new selling price: $$Selling = 1.2 \times C = 1.2 \times 16666.67 = 20000$$ --- 10. **Problem 34 (a): Probability of drawing a red ball from 6 red, 3 white, 1 black ball.** - Total balls: $$6 + 3 + 1 = 10$$ - Probability(red): $$\frac{6}{10} = 0.6$$ 11. **Problem 34 (b): Probability of NOT drawing a black ball.** - Probability(not black): $$1 - \frac{1}{10} = \frac{9}{10} = 0.9$$