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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You're on a roll — here's the solution to question 18: Part (a): Calculate the volume of the solid correct to two decimal places. The composite solid consists of a hemisphere and a frustum of a cone. Step 1: Identify the dimensions of the hemisphere. The diameter of the hemisphere is 30 cm. The radius of the hemisphere, R_H = (30)/(2) = 15 cm. Step 2: Calculate the volume of the hemisphere. The formula for the volume of a hemisphere is V_H = (2)/(3) R_H^3. Using = (22)/(7): V_H = (2)/(3) × (22)/(7) × (15)^3 V_H = (2)/(3) × (22)/(7) × 3375 V_H = 2 × (22)/(7) × 1125 V_H = (44 × 1125)/(7) V_H = (49500)/(7) cm^3 Step 3: Identify the dimensions of the frustum. The frustum is mounted on the hemisphere, so its bottom diameter is 30 cm. The radius of the bottom of the frustum, R_B = (30)/(2) = 15 cm. The radius of the top of the frustum, R_T = 10 cm. The entire solid has a vertical height of 60 cm. The height of the hemisphere is equal to its radius, R_H = 15 cm. The height of the frustum, h_F = Total height - Height of hemisphere h_F = 60 - 15 = 45 cm Step 4: Calculate the volume of the frustum. The formula for the volume of a frustum is V_F = (1)/(3) h_F (R_B^2 + R_B R_T + R_T^2). Using = (22)/(7): V_F = (1)/(3) × (22)/(7) × 45 × (15^2 + 15 × 10 + 10^2) V_F = (1)/(3) × (22)/(7) × 45 × (225 + 150 + 100) V_F = (22)/(7) × 15 × (475) V_F = (330 × 475)/(7) V_F = (156750)/(7) cm^3 Step 5: Calculate the total volume of the solid. The total volume V_total = V_H + V_F. V_total = (49500)/(7) + (156750)/(7) V_total = (49500 + 156750)/(7) V_total = (206250)/(7) V_total ≈ 29464.2857 cm^3 Rounding to two decimal places: The volume of the solid is 29464.29 cm^3. Part (b): The solid is melted and recast into cubes of side 16 cm. How many such cubes are realized? Step 1: Calculate the volume of one cube. The side of the cube, s = 16 cm. The volume of one cube, V_C = s^3. V_C = (16)^3 V_C = 16 × 16 × 16 V_C = 4096 cm^3 Step 2: Calculate the number of cubes. Number of cubes = Total volume of solidVolume of one cube Number of cubes = (206250)/(7)4096 Number of cubes = (206250)/(7 × 4096) Number of cubes = (206250)/(28672) Number of cubes ≈ 7.1933 Since only whole cubes can be realized, we take the integer part. The number of cubes realized is 7. Got more? Send 'em!