the Final Exam Question 45, 8.2 .63 Points: 0 of 1 To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars, the Detta, the Beta, and the Sigma Each Deita requires 13 hours for painting. 3 hours for drying, and 5 hours for polishing. A Beta requires 33 hours for painting, 8 hours for dyying, and 6 hours for polishing, and a Sigma requires 8 hours for painting, 1 hour for drying, and 1 hour for polish If the company has 331 hours for painting, 69 hours for drying, and 71 hours for polishing per month, how many of each type of car are produced? Select the correct choice below and fill in any answer boxes within your choice. A. The Epsilion Motor Company produces \( \square \) Deitas, \( \square \) Betas, and \( \square \) Sigmas in a month. (Simplify your answers.) B. There are infinitely many combinations of each type of car that can be produced. Using ordered triplets, the solution can be written as \( ((\mathrm{P}, \mathrm{B}, \mathrm{S}) / \mathrm{D}=\mathrm{D}, \mathrm{B}=\mathrm{D}, \mathrm{S} \) any real number). \( \square \) \( \square \) (Simplify your answers. Type expressions using \( S \) as the variable as needed.) C. There is no possible combination of each type of car that can be produced that meets the criteria.

1. Write the equations based on the resource requirements: - Painting: \(13D + 33B + 8S = 331\) - Drying: \(3D + 8B + S = 69\) - Polishing: \(5D + 6B + S = 71\) 2. Subtract the drying equation from the polishing equation to eliminate \(S\): \[ (5D + 6B + S) - (3D + 8B + S) = 71 - 69 \] \[ 2D - 2B = 2 \quad \Rightarrow \quad D - B = 1 \] Thus, \[ D = B + 1 \] 3. Substitute \(D = B + 1\) into the drying equation: \[ 3(B+1) + 8B + S = 69 \] \[ 3B + 3 + 8B + S = 69 \quad \Rightarrow \quad 11B + S = 66 \] Solve for \(S\): \[ S = 66 - 11B \] 4. Substitute \(D = B+1\) and \(S = 66 - 11B\) into the painting equation: \[ 13(B+1) + 33B + 8(66-11B) = 331 \] Expand and simplify: \[ 13B + 13 + 33B + 528 - 88B = 331 \] Combine like terms: \[ (13B + 33B - 88B) + (13 + 528) = 331 \] \[ -42B + 541 = 331 \] Solve for \(B\): \[ -42B = 331 - 541 \quad \Rightarrow \quad -42B = -210 \quad \Rightarrow \quad B = 5 \] 5. Find \(D\) and \(S\): \[ D = B + 1 = 5 + 1 = 6 \] \[ S = 66 - 11B = 66 - 55 = 11 \] 6. Verify the solution in each equation: - Painting: \(13(6) + 33(5) + 8(11) = 78 + 165 + 88 = 331\) - Drying: \(3(6) + 8(5) + 11 = 18 + 40 + 11 = 69\) - Polishing: \(5(6) + 6(5) + 11 = 30 + 30 + 11 = 71\) The Epsilon Motor Company produces \(6\) Deitas, \(5\) Betas, and \(11\) Sigmas in a month.