A company produces two products, x and y, and the cost associated with producing these is modeled by the function \(C(x, y) = x^2 + 2y^2 – 4x – 8y + 20\). Due to resource constraints, the total production is fixed so that \(y = 10 – x\).
(a) Express the cost function solely in terms of \(x\) by substituting the constraint.
(b) Differentiate the resulting function to find the critical point. Determine the value of \(x\) that minimizes the cost and compute the corresponding minimum cost.
(c) Interpret the result in the context of production optimization, explaining what the optimal production quantities imply.
Question
Substitute \(y = 10 – x\) into the cost function \(C(x, y) = x^2 + 2y^2 – 4x – 8y + 20\) to express it solely in terms of \(x\). Simplify your answer.
Hint
Hint
Hint
Substitute the constraint carefully and expand the squared term correctly.
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- Formula Booklet
Complete Answer
\(C(x) = 3x^2 – 36x + 140\)
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Question
Differentiate the function \(C(x) = 3x^2 – 36x + 140\) to find its critical point. Determine the value of \(x\) that minimizes the cost and compute the minimum cost.
Hint
Hint
Hint
Differentiate \(C(x)\) and set the derivative equal to zero to solve for \(x\).
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- Formula Booklet
Complete Answer
\(x = 6, \; C(6) = 32\)
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Question
Interpret the results from parts (a) and (b) in the context of production optimization. Explain what the optimal values indicate about the production quantities of products x and y.
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Hint
Hint
Relate the value of \(x\) and the constraint \(y = 10-x\) to the optimal production quantities.
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- Formula Booklet
Complete Answer
Optimal production is achieved when \(x = 6\) and \(y = 4\), meaning that producing 6 units of product x and 4 units of product y minimizes the cost.
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