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.
ScanSolve AI Answer
Hey Rigan, good to see you again. Here are the solutions for the solid Q: a) Find the slant height of the cone correct to 2 decimal places. Step 1: Identify the formula for the slant height of a cone. The slant height l of a cone is given by the Pythagorean theorem, where r is the radius and h_cone is the height of the cone. l = sqrt(r^2 + h_cone)^2 Step 2: Substitute the given values into the formula. Given r = 2.1 m and h_cone = 5 m. l = sqrt((2.1)^2 + (5)^2) l = sqrt(4.41 + 25) l = sqrt(29.41) Step 3: Calculate the slant height and round to 2 decimal places. l ≈ 5.4231 m l ≈ 5.42 m b) Find the total surface area of the solid Q correct to 4 significant figures. Step 1: Identify the components of the total surface area. The total surface area (TSA) of the solid Q is the sum of the surface area of the hemisphere, the lateral surface area of the cylinder, and the lateral surface area of the cone. TSA = (Area of hemisphere) + (Lateral area of cylinder) + (Lateral area of cone) TSA = 2 r^2 + 2 r h_c + r l We can factor out r: TSA = r (2r + 2h_c + l) Step 2: Substitute the given values and the calculated slant height into the formula. Given r = 2.1 m, h_c = 14 m, = (22)/(7), and l ≈ 5.4231 m (using the unrounded value for accuracy). TSA = (22)/(7) × 2.1 × (2 × 2.1 + 2 × 14 + 5.4231) TSA = 6.6 × (4.2 + 28 + 5.4231) TSA = 6.6 × (37.6231) Step 3: Calculate the total surface area and round to 4 significant figures. TSA ≈ 248.31246 m^2 TSA ≈ 248.3 m^2 c) Find the volume of solid Q to the nearest metres. Step 1: Identify the components of the total volume. The total volume (V) of the solid Q is the sum of the volume of the hemisphere, the volume of the cylinder, and the volume of the cone. V = (Volume of hemisphere) + (Volume of cylinder) + (Volume of cone) V = (2)/(3) r^3 + r^2 h_c + (1)/(3) r^2 h_cone We can factor out r^2: V = r^2 ((2)/(3)r + h_c + (1)/(3)h_cone) Step 2: Substitute the given values into the formula. Given r = 2.1 m, h_c = 14 m, h_cone = 5 m, and = (22)/(7). V = (22)/(7) × (2.1)^2 × ((2)/(3) × 2.1 + 14 + (1)/(3) × 5) V = (22)/(7) × 4.41 × (1.4 + 14 + (5)/(3)) V = 13.86 × (15.4 + 1.6666...) V = 13.86 × (17.0666...) Step 3: Calculate the total volume and round to the nearest metre. V ≈ 236.5824 m^3 V ≈ 237 m^3 Send me the next one 📸