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's the solution to the problem: a) Height of the solid Step 1: Identify the components of the solid and their dimensions. The solid consists of a conical part, a cylindrical part, and a hemispherical part. Diameter (D) = 10 cm, so radius (r) = (10)/(2) = 5 cm. Height of cylindrical part (h_c) = 15 cm. Slant height of conical part (l) = 13 cm. The height of the hemispherical part is equal to its radius, r = 5 cm. Step 2: Calculate the height of the conical part (h_k). Using the Pythagorean theorem for the cone: l^2 = r^2 + h_k^2. 13^2 = 5^2 + h_k^2 169 = 25 + h_k^2 h_k^2 = 169 - 25 h_k^2 = 144 h_k = sqrt(144) = 12 cm Step 3: Calculate the total height of the solid. Total height (H) = height of conical part + height of cylindrical part + height of hemispherical part. H = h_k + h_c + r H = 12 cm + 15 cm + 5 cm H = 32 cm The height of the solid is 32 cm. b) Volume of the solid, correct to 1 decimal place Step 1: Calculate the volume of the conical part (V_k). The formula for the volume of a cone is V_k = (1)/(3) r^2 h_k. Use = 3.142. V_k = (1)/(3) × 3.142 × (5 cm)^2 × 12 cm V_k = (1)/(3) × 3.142 × 25 cm^2 × 12 cm V_k = 3.142 × 25 cm^2 × 4 cm V_k = 314.2 cm^3 Step 2: Calculate the volume of the cylindrical part (V_c). The formula for the volume of a cylinder is V_c = r^2 h_c. V_c = 3.142 × (5 cm)^2 × 15 cm V_c = 3.142 × 25 cm^2 × 15 cm V_c = 1178.25 cm^3 Step 3: Calculate the volume of the hemispherical part (V_h). The formula for the volume of a hemisphere is V_h = (2)/(3) r^3. V_h = (2)/(3) × 3.142 × (5 cm)^3 V_h = (2)/(3) × 3.142 × 125 cm^3 V_h = (785.5)/(3) cm^3 V_h ≈ 261.833 cm^3 Step 4: Calculate the total volume of the solid. Total volume (V) = V_k + V_c + V_h. V = 314.2 cm^3 + 1178.25 cm^3 + 261.833 cm^3 V = 1754.283 cm^3 Rounding to 1 decimal place: V ≈ 1754.3 cm^3 The volume of the solid is 1754.3 cm^3. c) Surface area of the solid, correct to 1 decimal place Step 1: Calculate the curved surface area of the conical part (A_k). The formula for the curved surface area of a cone is A_k = r l. A_k = 3.142 × 5 cm × 13 cm A_k = 3.142 × 65 cm^2 A_k = 204.23 cm^2 Step 2: Calculate the curved surface area of the cylindrical part (A_c). The formula for the curved surface area of a cylinder is A_c = 2 r h_c. A_c = 2 × 3.142 × 5 cm × 15 cm A_c = 3.142 × 150 cm^2 A_c = 471.3 cm^2 Step 3: Calculate the curved surface area of the hemispherical part (A_h). The formula for the curved surface area of a hemisphere is A_h = 2 r^2. A_h = 2 × 3.142 × (5 cm)^2 A_h = 2 × 3.142 × 25 cm^2 A_h = 3.142 × 50 cm^2 A_h = 157.1 cm^2 Step 4: Calculate the total surface area of the solid. Total surface area (A) = A_k + A_c + A_h. A = 204.23 cm^2 + 471.3 cm^2 + 157.1 cm^2 A = 832.63 cm^2 Rounding to 1 decimal place: A ≈ 832.6 cm^2 The surface area of the solid is 832.6 cm^2. That's 2 down. 3 left today — send the next one.