Explanation Given information: Sugar refinery can produce x tons of sugar per week at a weekly cost of 0.1 x 2 + 5 x + 2250 Explanation: Let C ( x ) be the cost function. Hence, C ( x ) = 0.1 x 2 + 5 x + 2250 Let A ( x ) be the average cost function. The formula for the average cost is, A ( x ) = C ( x ) x ⇒ A ( x ) = 0.1 x 2 + 5 x + 2250 x . Marginal cost is the first derivative of cost function. Therefore, the marginal cost = C ' ( x ) = 0.2 x + 5 . Find the first and the second derivative of average cost. By using the quotient rule of differentiation, ⇒ A ' ( x ) = x ⋅ C ' ( x ) − C ( x ) ⋅ 1 x 2 ⇒ A ' ( x ) = x ⋅ ( 0.2 x + 5 ) − 0.1 x 2 + 5 x + 2250 x 2 ⇒ A ' ( x ) = 0.1 x 2 − 2250 x 2 Now, we find the second derivative using quotient rule of differentiation, ⇒ A ' ' ( x ) = x 2 ( 0.2 x ) − ( 0.1 x 2 − 2250 ) ( 2 x ) x 4 ⇒ A ' ' ( x ) = 0.2 x 3 − 0.2 x 3 − 4500 x x 4 ⇒ A ' ' ( x ) = − 4500 x 3 To find the critical point, set A ' ( x ) = 0 0

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Chapter 3.1, Problem 47E

To determine

The level of production for which the average cost is at a minimum and show that the average cost equals the marginal cost at that level of production, where the sugar refinery can produce x tons of sugar per week at a weekly cost of 0.1x2 + 5x + 2250

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Chapter 3.1, Problem 46E

Chapter 3.1, Problem 48E

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Calculus & Its Applications (14th Edition)

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The level of production for which the average cost is at a minimum and show that the average cost equals the marginal cost at that level of production, where the sugar refinery can produce x tons of sugar per week at a weekly cost of 0.1 x 2 + 5 x + 2250

Solution Summary: The author explains that the average cost equals the marginal cost at the level of production. Marginal cost is the first derivative of cost function.

The level of production for which the average cost is at a minimum and show that the average cost equals the marginal cost at that level of production, where the sugar refinery can produce x tons of sugar per week at a weekly cost of 0.1x2 + 5x + 2250

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Chapter 3 Solutions