A manufacturing company has collected cost data (in thousand dollars) based on production units (u) as follows:
– For 100 units, the cost is 60.
– For 200 units, the cost is 100.
– For 300 units, the cost is 180.
Using this data, answer the following:
(a) Determine a quadratic polynomial cost function of the form \(C(u)= au^2 + bu + c\) that fits the data.
(b) Differentiate the cost function you obtained to find the marginal cost function \(C'(u)\).
(c) Use the marginal cost function to estimate the approximate change in cost when production increases from 300 to 301 units.
Question
Determine a quadratic polynomial cost function of the form \(C(u)= au^2 + bu + c\) that fits the following data:
\(u = 100,\; C(100)= 60\)
\(u = 200,\; C(200)= 100\)
\(u = 300,\; C(300)= 180\)
Hint
Hint
Hint
Set up a system of equations for \(a\), \(b\), and \(c\) using the given data points.
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\(a = 0.002,\; b = -0.2,\; c = 60\)
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Question
Differentiate the cost function \(C(u)= 0.002u^2 – 0.2u + 60\) to find the marginal cost function \(C'(u)\).
Hint
Hint
Hint
Differentiate each term using the power rule.
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\(C'(u)= 0.004u – 0.2\)
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Question
Using the marginal cost function \(C'(u)= 0.004u – 0.2\), estimate the approximate change in cost when production increases from 300 to 301 units.
Hint
Hint
Hint
Evaluate the derivative at \(u=300\) to approximate the cost increase per additional unit.
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Approximately \(1.0\) (thousand dollars)
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