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
a) Step 1: Calculate the calibration factor. The calibration factor is double the square root of 16. Calibration factor = 2 × sqrt(16) = 2 × 4 = 8 Step 2: Calculate the actual particle size for each number of divisions. The actual particle size is found by multiplying the number of divisions by the calibration factor (8). | No of division | 4 | 5 | 6 | 7 | 8 | 10 | 12 | 14 | 16 | 17 | 18 | 19 | 20 | 22 | 23 | 24 | 25 | | :------------- | :- | :- | :- | :- | :- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | :-- | | Actual particle size | 32 | 40 | 48 | 56 | 64 | 80 | 96 | 112 | 128 | 136 | 144 | 152 | 160 | 176 | 184 | 192 | 200 | The calibration factor is 8. b) Step 1: Determine the mid-class (X) for each particle size range. For 0-50: X = (0+50)/(2) = 25 For 51-100: X = (51+100)/(2) = 75.5 For 101-150: X = (101+150)/(2) = 125.5 For 151-200: X = (151+200)/(2) = 175.5 Step 2: Determine the frequency (F) for each class by counting the actual particle sizes from part (a) that fall into each range. The actual particle sizes are: 32, 40, 48, 56, 64, 80, 96, 112, 128, 136, 144, 152, 160, 176, 184, 192, 200. 0-50: 32, 40, 48 F = 3 51-100: 56, 64, 80, 96 F = 4 101-150: 112, 128, 136, 144 F = 4 151-200: 152, 160, 176, 184, 192, 200 F = 6 Step 3: Calculate Frequency × mid class (FX) for each class. | Particle size | Mid class (X) | Frequency (F) | Frequency X mid class (FX) | | :------------ | :------------ | :------------ | :------------------------- | | 0-50 | 25 | 3 | 25 × 3 = 75 | | 51-100 | 75.5 | 4 | 75.5 × 4 = 302 | | 101-150 | 125.5 | 4 | 125.5 × 4 = 502 | | 151-200 | 175.5 | 6 | 175.5 × 6 = 1053 | Step 4: Calculate the sum of frequencies ( F) and the sum of (FX) ( FX). F = 3 + 4 + 4 + 6 = 17 FX = 75 + 302 + 502 + 1053 = 1932 Step 5: Calculate the mean particle size using the formula ( FX)/( F). Mean particle size = (1932)/(17) ≈ 113.65 The mean particle size is 113.65. 3 done, 2 left today. You're making progress.