1. **Problem statement:** Find the areas related to a frustum of a cone and a hemisphere, and solve related algebraic factorization and simplification problems. (a)(i) Find the area of the circular base of the frustum. The formula for the area of a circle is: $$A = \pi r^2$$ where $r$ is the radius. Given $r = 3.5$ cm, $$A = \pi (3.5)^2 = \pi \times 12.25 = 12.25\pi \approx 38.48 \text{ cm}^2$$ (a)(ii) Find the curved surface area of the frustum. The curved surface area (CSA) of a frustum is given by: $$\text{CSA} = \pi (R + r) l$$ where $R = 4.2$ cm, $r = 3.5$ cm, and $l = 8$ cm. Calculate: $$\text{CSA} = \pi (4.2 + 3.5) \times 8 = \pi \times 7.7 \times 8 = 61.6\pi \approx 193.39 \text{ cm}^2$$ (a)(iii) Find the surface area of the hemisphere. The surface area of a hemisphere is: $$2\pi r^2$$ Using $r = 3.5$ cm, $$2\pi (3.5)^2 = 2\pi \times 12.25 = 24.5\pi \approx 76.97 \text{ cm}^2$$ (b) A similar solid has a total area of 81.51 cm². Determine the radius of its base. Since the solids are similar, areas scale by the square of the scale factor $k^2$. Let original total area $A_1$ be sum of parts from (a): $$A_1 = 38.48 + 193.39 + 76.97 = 308.84 \text{ cm}^2$$ Let new total area $A_2 = 81.51$ cm². Scale factor squared: $$k^2 = \frac{A_2}{A_1} = \frac{81.51}{308.84} \approx 0.264$$ Scale factor: $$k = \sqrt{0.264} \approx 0.514$$ Original base radius $r = 3.5$ cm, so new radius: $$r_{new} = k \times 3.5 = 0.514 \times 3.5 \approx 1.80 \text{ cm}$$ 16. Factorize completely $3x^2 - 2xy - y^2$. Look for factors of $3 \times (-1) = -3$ that sum to $-2$: $$-3 \text{ and } 1$$ Rewrite middle term: $$3x^2 - 3xy + xy - y^2$$ Group: $$(3x^2 - 3xy) + (xy - y^2) = 3x(x - y) + y(x - y)$$ Factor out common: $$(3x + y)(x - y)$$ 17. Factorize $a^2 - b^2$. This is a difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ Find exact value of $2557^2 - 2547^2$ using difference of squares: Let $a = 2557$, $b = 2547$. $$2557^2 - 2547^2 = (2557 - 2547)(2557 + 2547) = 10 \times 5104 = 51040$$ 18. Simplify the expression: $$\frac{3a^2 + 4ab + b^2}{4a^2 + 3ab - b^2}$$ Factor numerator: Try to factor $3a^2 + 4ab + b^2$. Check if it factors as $(pa + qb)(ra + sb)$: Try $(3a + b)(a + b) = 3a^2 + 3ab + ab + b^2 = 3a^2 + 4ab + b^2$ correct. Factor denominator: Try $(4a^2 + 3ab - b^2)$. Try $(4a - b)(a + b) = 4a^2 + 4ab - ab - b^2 = 4a^2 + 3ab - b^2$ correct. So expression becomes: $$\frac{(3a + b)(a + b)}{(4a - b)(a + b)}$$ Cancel common factor $(a + b)$: $$\frac{\cancel{(a + b)}(3a + b)}{(4a - b)\cancel{(a + b)}} = \frac{3a + b}{4a - b}$$ 19. Four farmers took their goats to market. Mohamed had two more goats than Ali. Koech had 3 times as many goats as Mohamed. Odupoy had 10 goats less than both Mohamed and Koech combined. Let Ali have $x$ goats. Mohamed has $x + 2$ goats. Koech has $3(x + 2) = 3x + 6$ goats. Odupoy has $(x + 2) + (3x + 6) - 10 = 4x - 2$ goats. This completes the problem setup.