Explanation Given Information: The daily cost to the firm of reduction is C ( q ) = 4000 + 100 q 2 . Where, q is the reduction in emissions. Formula used: Steps to solve optimization problems: Step 1: Identify the unknowns with the aid of diagram. Step 2: Identify the objective function, the quantity to maximize or minimize. Step 3: Identify the constraint, the equations relating variables or inequalities expressing limitations on the value of the variables. Step 4: State the optimization problem Step 5: Eliminate the extra variables, solve the constraint for one unknown and substitute in the objective function. Step 6: Find the absolute maximum or absolute minimum of the objective function. Calculation: As it is provided that the daily cost to the firm of reduction is: C ( q ) = 4000 + 100 q 2 Where, q is the reduction in emissions. Step 1: Identify the unknowns There is one unknown q which is the number reduction in emissions. Step 2: Identify the objective functions: The average cost C ¯ ( q ) is given by: C ( q ) q = 4000 + 100 q 2 q C ¯ ( q ) = 4000 q + 100 q Step 3: Identify the constraint: C ¯ ( 0 ) is not defined. Step 4: State the optimization problem: The given objective function is given by: Minimize the objective function C ¯ ( q ) = 4000 q + 100 q Step 5: State and solve the given optimization problem: The objective function is given by: C ¯ ( q ) = 4000 q + 100 q Differentiating both sides of the equation to get: C ¯ ′ ( q ) = − 4000 q 2 + 100 Locating stationary points to get: C ¯ ′ ( q ) = 0 − 4000 q 2 + 100 = 0 Subtracting 100 from both sides to get: − 4000 q 2 + 100 − 100 = − 100 − 4000 q 2 = 100 Dividing both sides by − 1 to get: − 4000 q 2 1 ( − 1 ) = − 100 − 1 4000 q 2 = 100 Cross multiplying to get: 4000 100 = q 2 40 = q 2 Taking square root of 40 as 6

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Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.2, Problem 15E

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To calculate: The minimization of average cost per pound of pollutant modeled by the equation C(q)=4000+100q2 where, C is the daily cost to the firm of reduction and q is the reduction in emissions.

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The minimization of average cost per pound of pollutant modeled by the equation C ( q ) = 4000 + 100 q 2 where, C is the daily cost to the firm of reduction and q is the reduction in emissions.

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To calculate: The minimization of average cost per pound of pollutant modeled by the equation C(q)=4000+100q2 where, C is the daily cost to the firm of reduction and q is the reduction in emissions.

Chapter 12 Solutions