1. **Problem 1: Calculate the total surface area of the large solid** The two solids are mathematically similar. Given: - Volume of large solid $V_L = 416$ cm³ - Volume of small solid $V_S = 52$ cm³ - Total surface area of small solid $A_S = 60$ cm² We need to find the total surface area of the large solid $A_L$. 2. **Formula and rules:** For similar solids, the ratio of volumes is the cube of the scale factor $k$: $$\frac{V_L}{V_S} = k^3$$ The ratio of surface areas is the square of the scale factor: $$\frac{A_L}{A_S} = k^2$$ 3. **Calculate the scale factor $k$:** $$k^3 = \frac{V_L}{V_S} = \frac{416}{52} = 8$$ So, $$k = \sqrt[3]{8} = 2$$ 4. **Calculate the total surface area of the large solid:** $$\frac{A_L}{60} = 2^2 = 4$$ Therefore, $$A_L = 60 \times 4 = 240 \text{ cm}^2$$ --- 5. **Problem 2: Probability that two pencils chosen are different colours** Given: - Total pencils = 13 - Red = 4 - Green = 7 - Yellow = 2 Two pencils are chosen without replacement. 6. **Calculate total number of ways to choose 2 pencils:** $$\binom{13}{2} = \frac{13 \times 12}{2} = 78$$ 7. **Calculate number of ways to choose 2 pencils of the same colour:** - Red: $\binom{4}{2} = 6$ - Green: $\binom{7}{2} = 21$ - Yellow: $\binom{2}{2} = 1$ Total same colour pairs = $6 + 21 + 1 = 28$ 8. **Calculate number of ways to choose 2 pencils of different colours:** $$78 - 28 = 50$$ 9. **Calculate the probability:** $$P = \frac{50}{78} = \frac{25}{39}$$ --- 10. **Problem 3: Complete the table for $y = -2x^2 + x + 3$** Given table: | x | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | | y | -7 | -3 | | 2 | 3 | | 0 | | | -7 | 11. **Calculate missing values:** - For $x = -1$: $$y = -2(-1)^2 + (-1) + 3 = -2(1) -1 + 3 = -2 -1 + 3 = 0$$ - For $x = 0.5$: $$y = -2(0.5)^2 + 0.5 + 3 = -2(0.25) + 0.5 + 3 = -0.5 + 0.5 + 3 = 3$$ - For $x = 1.5$: $$y = -2(1.5)^2 + 1.5 + 3 = -2(2.25) + 1.5 + 3 = -4.5 + 1.5 + 3 = 0$$ - For $x = 2$: $$y = -2(2)^2 + 2 + 3 = -2(4) + 2 + 3 = -8 + 2 + 3 = -3$$ 12. **Completed table:** | x | -2 | -1.5 | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | | y | -7 | -3 | 0 | 2 | 3 | 3 | 0 | 0 | -3 | -7 |