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 complete solution for Question 2: QUESTION 2 The table provides data for "Quantity of lubricant x 10^3 gallons" and "Total machine hours x 10^4 hrs". Let Y be the quantity of lubricant in 10^3 gallons and X be the total machine hours in 10^4 hrs. | Period | Y (10^3 gallons) | X (10^4 hrs) | XY | X^2 | |---|---|---|---|---| | 2024 | 29 | 15 | 435 | 225 | | 2023 | 23 | 10 | 230 | 100 | | 2022 | 16 | 6 | 96 | 36 | | 2021 | 11 | 4 | 44 | 16 | | 2020 | 5 | 2 | 10 | 4 | | Sum | Y = 84 | X = 37 | XY = 815 | X^2 = 381 | n = 5 i) Estimate the quantity of lubricant to be required in Year 2025 when the machines are planned to work for a total of 8 × 10^5 Hours. Step 1: Calculate the slope (b) for the linear regression equation Y = a + bX. b = (n( XY) - ( X)( Y))/(n( X^2) - ( X)^2) b = (5(815) - (37)(84))/(5(381) - (37)^2) b = (4075 - 3108)/(1905 - 1369) b = (967)/(536) ≈ 1.8041 Step 2: Calculate the y-intercept (a) for the linear regression equation. a = ( Y - b( X))/(n) a = (84 - 1.8041(37))/(5) a = (84 - 66.7517)/(5) a = (17.2483)/(5) ≈ 3.4497 Step 3: Formulate the regression equation and determine the value of X for 2025. The regression equation is Y = 3.4497 + 1.8041X. The total machine hours for 2025 are 8 × 10^5 hours. Since X is in 10^4 hours, we convert: X = 8 × 10^5 hours10^4 hours/unit = 80 Step 4: Estimate the quantity of lubricant (Y) for 2025. Y = 3.4497 + 1.8041(80) Y = 3.4497 + 144.328 Y = 147.7777 Since Y is in 10^3 gallons, the quantity of lubricant is 147.7777 × 10^3 gallons. The estimated quantity of lubricant required in Year 2025 is 147.78 × 10^3 gallons. ii) For the 2026 & beyond, the lubricating oil consumption is to be pegged at 15 × 10^3 gallons per annum. What is the expected total production hour for the machine per annum for the period? Step 1: Set the lubricant consumption (Y) to the given value. The lubricating oil consumption is 15 × 10^3 gallons, so Y = 15 in our units. Step 2: Use the regression equation to find the corresponding total machine hours (X). 15 = 3.4497 + 1.8041X 15 - 3.4497 = 1.8041X 11.5503 = 1.8041X X = (11.5503)/(1.8041) ≈ 6.4022 Step 3: Convert X back to total production hours. Since X is in 10^4 hrs, the total production hour is 6.4022 × 10^4 hrs. The expected total production hour is 6.40 × 10^4 hours. iii) How many oilers will be required in year 2025? If each Oiler works at the rate of 100 gallons/hour. Assume 250 working day per annum & 8 hours per day. Use factor for Oiler = 0.8. Step 1: Determine the total lubricant required in 2025. From part (i), the total lubricant required in 2025 is 147.7777 × 10^3 gallons, which is 147,777.7 gallons. Step 2: Calculate the total effective working hours for one Oiler per annum. Total working days per annum = 250 days Hours per day = 8 hours Total hours per Oiler per annum = 250 days/year × 8 hours/day = 2000 hours/year. Step 3: Calculate the effective lubricant handling capacity of one Oiler per annum. Rate of work for one Oiler = 100 gallons/hour. Factor for Oiler = 0.8. Effective rate of work = 100 gallons/hour × 0.8 = 80 gallons/hour. Effective gallons an Oiler can handle per annum = 2000 hours/year × 80 gallons/hour = 160,000 gallons/year. Step 4: Calculate the number of oilers required. Number of oilers = Total lubricant requiredEffective gallons per Oiler per annum Number of oilers = 147,777.7 gallons160,000 gallons/oiler Number of oilers ≈ 0.9236 Since you cannot have a fraction of an Oiler, we round up to the nearest whole number. The number of oilers required in year 2025 is 1 Oiler. Send me the next one 📸