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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Step 1: Define variables and set up initial relationships. Let x be the cost of producing 1 kg of chocolate bars in factory A. Let y be the cost of producing 1 kg of eclairs in factory A. The cost of 1 kg of chocolate bars in factory B is (6)/(5) times the cost in factory A, so it is (6)/(5)x. The cost of 1 kg of eclairs in factory B is (4)/(5) times the cost in factory A, so it is (4)/(5)y. a) Form two simplified simultaneous equations representing this information. From the problem statement: 1. It costs Kshs 460,000 to produce 1 tonne (1000 kg) of chocolate bars and 800 kg of eclairs in factory A. 1000x + 800y = 460,000 Divide by 100: 10x + 8y = 4600 Divide by 2: 5x + 4y = 2300 (Equation 1) 2. It costs Kshs 534,000 to produce the same quantities (1000 kg of chocolate bars and 800 kg of eclairs) in factory B. Cost of 1000 kg chocolate bars in factory B: 1000 × (6)/(5)x = 1200x. Cost of 800 kg eclairs in factory B: 800 × (4)/(5)y = 640y. 1200x + 640y = 534,000 Divide by 20: 60x + 32y = 26,700 Divide by 4: 15x + 8y = 6675 (Equation 2) The two simplified simultaneous equations are: 5x + 4y &= 2300 \\ 15x + 8y &= 6675 b) Use matrix method to find the cost of producing 1 kg of chocolate bars and 1 kg of eclairs in factory A. The system of equations from part (a) is: 5x + 4y &= 2300 \\ 15x + 8y &= 6675 Write this system in matrix form AX = B: 5 & 4 \\ 15 & 8 x \\ y = 2300 \\ 6675 Find the determinant of matrix A: (A) = (5 × 8) - (4 × 15) = 40 - 60 = -20 Find the inverse of matrix A: A^-1 = (1)/((A)) 8 & -4 \\ -15 & 5 = (1)/(-20) 8 & -4 \\ -15 & 5 Now, calculate X = A^-1B: x \\ y = -(1)/(20) 8 & -4 \\ -15 & 5 2300 \\ 6675 x \\ y = -(1)/(20) (8 × 2300) + (-4 × 6675) \\ (-15 × 2300) + (5 × 6675) x \\ y = -(1)/(20) 18400 - 26700 \\ -34500 + 33375 x \\ y = -(1)/(20) -8300 \\ -1125 x \\ y = (-8300)/(-20) \\ (-1125)/(-20) = 415 \\ 56.25 So, x = 415 and y = 56.25. The cost of producing 1 kg of chocolate bars in factory A is Kshs 415. The cost of producing 1 kg of eclairs in factory A is Kshs 56.25. 1 kg chocolate bars: Kshs 415, 1 kg eclairs: Kshs 56.25 c) Find the cost of producing 100 kg of chocolate bars and 50 kg of eclairs in factory B. First, find the cost per kg in factory B: Cost of 1 kg of chocolate bars in factory B = (6)/(5)x = (6)/(5) × 415 = 6 × 83 = 498 Kshs. Cost of 1 kg of eclairs in factory B = (4)/(5)y = (4)/(5) × 56.25 = 4 × 11.25 = 45 Kshs. Now, calculate the total cost for 100 kg of chocolate bars and 50 kg of eclairs in factory B: Total cost = (100 kg × 498 Kshs/kg) + (50 kg × 45 Kshs/kg) Total cost = 49800 + 2250 Total cost = 52050 Kshs. Kshs 52,050 3 done, 2 left today. You're making progress.