A manufacturer of widgets estimates that if her weekly factory production is q t

Question

A manufacturer of widgets estimates that if her weekly factory production is q thousand widgets, then the associated weekly revenue will be R(q) = 28q - 0.04q3 thousand dollars. The factory has a weekly capacity of 25,000 widgets, but current weekly widget production is only 10,000. The factory has $ 50,000 in fixed costs each week and each widget requires $ 14 of materials. Additionally, high production levels require substantial increases in machinery maintenance. The factory owner estimates that her weekly cost to produce q thousand widgets is C(q) = 50 + 14q + 0.2q2 thousand dollars. What is the marginal weekly cost at a production level of 10,000 widgets? What is the marginal weekly revenue at a production level of 10,000 widgets?

Explanation / Answer

A) Marginal weekly cost is given by the derivative of the cost function. C'(q) = 14+0.4q, where q represents the number of widgets in thousands, so for 10,000 widgets we have q = 10. Evaluating: C'(10) = 14+(0.4)(10) = 14+4 = 28. The units of C are thousands of dollars and the units of q are thousands of widgets, so this represents 28 $1000/1000 widgets or 28 $/widget. B) Similarly, marginal weekly revenue is given by the derivative of the weekly revenue function: R'(q) = 28 - 0.12q^2 and we need to evaluate this for q = 10, so R'(10) = 28-(0.12)(10)^2 = 28-(0.12)(100) = 28-12 = 16 $/widget.

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