This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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I will solve question 8 based on the image provided. a) Calculate the capacity of the bucket. b) Calculate the area of material required to make the bucket. The bucket is in the shape of a frustum of a cone. From the diagram: Top diameter = 50 cm, so top radius R = (50)/(2) = 25 cm. Bottom diameter = 8 cm, so bottom radius r = (8)/(2) = 4 cm. Vertical height h = 42.5 cm. Step 1: Calculate the capacity (volume) of the bucket. The formula for the volume of a frustum is V = (1)/(3) h (R^2 + Rr + r^2). Substitute the given values: R = 25 cm, r = 4 cm, h = 42.5 cm. V = (1)/(3) × × 42.5 × (25^2 + 25 × 4 + 4^2) V = (1)/(3) × × 42.5 × (625 + 100 + 16) V = (1)/(3) × × 42.5 × 741 V = × 42.5 × 247 V = 10497.5 cm^3 Using ≈ 3.14159: V ≈ 10497.5 × 3.14159 V ≈ 32979.6 cm^3 The capacity of the bucket is approximately 32979.6 cm^3. Step 2: Calculate the area of material required to make the bucket. This refers to the curved surface area of the frustum plus the area of the bottom circular base. The top is open. First, we need to find the slant height l of the frustum. The formula for the slant height of a frustum is l = sqrt(h^2 + (R-r)^2). l = sqrt(42.5^2 + (25-4)^2) l = sqrt(42.5^2 + 21^2) l = sqrt(1806.25 + 441) l = sqrt(2247.25) l ≈ 47.405 cm Step 3: Calculate the curved surface area of the frustum. The formula for the curved surface area of a frustum is A_curved = (R+r)l. A_curved = (25+4) × 47.405 A_curved = × 29 × 47.405 A_curved = 1374.745 cm^2 Using ≈ 3.14159: A_curved ≈ 1374.745 × 3.14159 A_curved ≈ 4319.9 cm^2 Step 4: Calculate the area of the bottom circular base. The formula for the area of a circle is A_base = r^2. A_base = × 4^2 A_base = 16 cm^2 Using ≈ 3.14159: A_base ≈ 16 × 3.14159 A_base ≈ 50.265 cm^2 Step 5: Calculate the total area of material required. Total area A_total = A_curved + A_base. A_total = 4319.9 + 50.265 A_total = 4370.165 cm^2 The area of material required to make the bucket is approximately 4370.2 cm^2. Drop the next question.