1. **Problem 1: Triangular face and pyramid heights** Since the figure is not fully described, we assume a triangular face of a pyramid with base and height given or implied. ii. Area of a triangular face formula: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ iii. Perpendicular height of a triangular face is the height used in the area formula. iv. Perpendicular height of the pyramid can be found using Pythagoras theorem if the slant height and half base are known: $$\text{height} = \sqrt{\text{slant height}^2 - \left(\frac{\text{base}}{2}\right)^2}$$ Represent the answer as a surd if it is irrational. 2. **Problem 2: Cone calculations** Given: Height $h = 20$ cm, radius $r = 14$ cm. i. Slant height $l$ of the cone: $$l = \sqrt{r^2 + h^2} = \sqrt{14^2 + 20^2} = \sqrt{196 + 400} = \sqrt{596} = 2\sqrt{149}$$ ii. Area of the base: $$\text{Base area} = \pi r^2 = \pi \times 14^2 = 196\pi$$ iii. Total surface area of the cone: $$\text{Total surface area} = \pi r^2 + \pi r l = \pi r (r + l) = \pi \times 14 (14 + 2\sqrt{149}) = 14\pi (14 + 2\sqrt{149})$$ 3. **Problem 3: Hemisphere and sphere surface areas** a) i. Total surface area of a hemisphere: $$\text{Curved surface area} = 2\pi a^2$$ $$\text{Base area} = \pi a^2$$ $$\text{Total surface area} = 2\pi a^2 + \pi a^2 = 3\pi a^2$$ ii. For radius $a=7$ cm: $$\text{Total surface area} = 3\pi \times 7^2 = 3\pi \times 49 = 147\pi \approx 461.81$$ b) i. Find radius $r$ of a sphere with surface area $154$ cm²: Surface area of sphere: $$4\pi r^2 = 154$$ $$r^2 = \frac{154}{4\pi} = \frac{77}{2\pi}$$ $$r = \sqrt{\frac{77}{2\pi}}$$ ii. External surface area of a solid cone with radius $0.75$ m (height not given, so only base area or lateral area if height known): If height $h$ is unknown, only base area: $$\text{Base area} = \pi r^2 = \pi \times 0.75^2 = 0.5625\pi$$ If height or slant height is given, lateral area can be calculated similarly. **Final answers:** - Slant height of cone: $2\sqrt{149}$ cm - Area of base of cone: $196\pi$ cm² - Total surface area of cone: $14\pi (14 + 2\sqrt{149})$ cm² - Total surface area of hemisphere radius 7 cm: $147\pi$ cm² - Radius of sphere with surface area 154 cm²: $\sqrt{\frac{77}{2\pi}}$ cm - External surface area of cone radius 0.75 m (base only): $0.5625\pi$ m²