A manufacturing firm produces three types of components: Component A, Component B, and Component C. The production manager gives the following constraints for the daily production plan:
– The sum of all components produced is 150 units.
– The number of units of Component A exceeds that of Component B by 10.
– The number of units of Component C exceeds that of Component B by 20.
Develop a system of linear equations based on these constraints and use it to determine the production quantity for each component.
Proceed with the following parts:
(a) Model the system of equations representing the constraints.
(b) Solve the system of equations to find the number of units for Components A, B, and C.
(c) Evaluate the solution in the context of production planning, discussing the feasibility of the production plan.
Question
Model the system of equations based on the following constraints:
– Total production: A + B + C = 150
– Component A exceeds Component B by 10: A – B = 10
– Component C exceeds Component B by 20: C – B = 20
Hint
Hint
Hint
Let A, B, and C represent the number of units of Component A, B, and C respectively. Translate each statement directly into an equation.
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Answer
Answer
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Explanation
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- Formula Booklet
Complete Answer
The system is:
1) A + B + C = 150
2) A – B = 10
3) C – B = 20
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Answer Explanation
Answer Explanation
Question
Solve the system of equations from part (a):
\(A + B + C = 150\)
\(A – B = 10\)
\(C – B = 20\)
Hint
Hint
Hint
Express A and C in terms of B using the second and third equations, then substitute these expressions into the first equation to solve for B.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
A = 50, B = 40, C = 60
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Question
Interpret the solution in the context of production planning. How does the solution reflect the feasibility of the production plan? Discuss any practical considerations for the manufacturing firm based on this solution.
Hint
Hint
Hint
Consider whether the quantities of components are realistic and if the production plan satisfies all initial constraints. Think about real-world factors that might affect the implementation of this plan.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
The solution suggests producing 50 units of Component A, 40 units of Component B, and 60 units of Component C. This plan is feasible as it satisfies all given constraints: the total production is 150 units (50+40+60=150), Component A production exceeds Component B by 10 units (50-40=10), and Component C production exceeds Component B by 20 units (60-40=20). Practically, these are positive integer quantities, which is realistic for production. However, the firm should also consider other factors not included in this model, such as demand for each component, production costs, resource availability, and storage capacity to ensure long-term feasibility and profitability.
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Answer Explanation
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