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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Here are the solutions to question 20. Question 20: Regular octagonal based prism Given: Side length of octagon, s = 10 cm Length (height) of prism, H = 50 cm a) The total surface area of the prism Step 1: Calculate the area of the regular octagonal base. The area of a regular octagon with side length s is given by the formula A_base = 2(1+sqrt(2))s^2. A_base = 2(1+sqrt(2))(10 cm)^2 A_base = 2(1+1.41421356)(100 cm^2) A_base = 2(2.41421356)(100 cm^2) A_base = 482.842712 cm^2 Step 2: Calculate the perimeter of the octagonal base. The perimeter of a regular octagon with side length s is P = 8s. P = 8 × 10 cm P = 80 cm Step 3: Calculate the lateral surface area of the prism. The lateral surface area is A_lateral = P × H. A_lateral = 80 cm × 50 cm A_lateral = 4000 cm^2 Step 4: Calculate the total surface area of the prism. The total surface area is A_total = 2 × A_base + A_lateral. A_total = 2 × 482.842712 cm^2 + 4000 cm^2 A_total = 965.685424 cm^2 + 4000 cm^2 A_total = 4965.685424 cm^2 Rounding to two decimal places: 4965.69 cm^2 b) The capacity of the prism in litres. Step 1: Calculate the volume of the prism. The volume of a prism is V = A_base × H. V = 482.842712 cm^2 × 50 cm V = 24142.1356 cm^3 Step 2: Convert the volume from cubic centimeters to litres. Since 1 litre = 1000 cm^3: V_litres = 24142.1356 cm^31000 cm^3/litre V_litres = 24.1421356 litres Rounding to two decimal places: 24.14 litres c) The radius of a cylinder that can exactly fill the prism. If the cylinder has a height of 28cm. Step 1: Equate the volume of the cylinder to the volume of the prism. "Exactly fill the prism" means their volumes are equal. V_cylinder = V_prism The volume of a cylinder is V_cylinder = r^2 h_cylinder. We have V_prism = 24142.1356 cm^3 and h_cylinder = 28 cm. r^2 (28 cm) = 24142.1356 cm^3 Step 2: Solve for the radius r. r^2 = 24142.1356 cm^3 × 28 cm r^2 = (24142.1356)/(87.964594) r^2 = 274.459069 cm^2 r = sqrt(274.459069 cm)^2 r = 16.566806 cm Rounding to two decimal places: 16.57 cm That's 2 down. 3 left today — send the next one.