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
Here is the solution for question 36. Step 1: Identify the given dimensions and formulas. The toy consists of a hemisphere surmounted by a cone. Height of the cone (h) = 2 cm. Diameter of the base = 4 cm. Radius (r) = diameter2 = 4 cm2 = 2 cm. We are given to use = (22)/(7). Step 2: Calculate the volume of the toy. The total volume of the toy is the sum of the volume of the hemisphere and the volume of the cone. Volume of hemisphere (V_hemisphere) = (2)/(3) r^3 Volume of cone (V_cone) = (1)/(3) r^2 h Calculate V_hemisphere: V_hemisphere = (2)/(3) × (22)/(7) × (2 cm)^3 V_hemisphere = (2)/(3) × (22)/(7) × 8 cm^3 V_hemisphere = (352)/(21) cm^3 Calculate V_cone: V_cone = (1)/(3) × (22)/(7) × (2 cm)^2 × 2 cm V_cone = (1)/(3) × (22)/(7) × 4 cm^2 × 2 cm V_cone = (176)/(21) cm^3 Calculate the total volume (V_toy): V_toy = V_hemisphere + V_cone V_toy = (352)/(21) cm^3 + (176)/(21) cm^3 V_toy = (352 + 176)/(21) cm^3 V_toy = (528)/(21) cm^3 Simplify the fraction by dividing numerator and denominator by 3: V_toy = (176)/(7) cm^3 The volume of the toy is (176)/(7) cm^3. Step 3: Calculate the surface area of the toy. The total surface area of the toy is the sum of the curved surface area of the hemisphere and the curved surface area of the cone. Curved surface area of hemisphere (CSA_hemisphere) = 2 r^2 Curved surface area of cone (CSA_cone) = r l, where l is the slant height of the cone. First, calculate the slant height (l) of the cone: l = sqrt(r^2 + h^2) l = sqrt((2 cm))^2 + (2 cm)^2 l = sqrt(4 cm)^2 + 4 cm^2 l = sqrt(8 cm)^2 l = 2sqrt(2) cm Calculate CSA_hemisphere: CSA_hemisphere = 2 × (22)/(7) × (2 cm)^2 CSA_hemisphere = 2 × (22)/(7) × 4 cm^2 CSA_hemisphere = (176)/(7) cm^2 Calculate CSA_cone: CSA_cone = (22)/(7) × 2 cm × 2sqrt(2) cm CSA_cone = 88sqrt(2)7 cm^2 Calculate the total surface area (SA_toy): SA_toy = CSA_hemisphere + CSA_cone SA_toy = (176)/(7) cm^2 + 88sqrt(2)7 cm^2 SA_toy = 176 + 88sqrt(2)7 cm^2 Factor out 88 from the numerator: SA_toy = 88(2 + sqrt(2))7 cm^2 The surface area of the toy is 88(2 + sqrt(2))7 cm^2. That's 2 down. 3 left today — send the next one.