Explanation Given Information: The provided equation is: P = x y z 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: Consider the provided equation: P = x y z Step 1: Identify the unknowns The unknowns are x and y . Step 2: Identify the objective functions: The objective function is given by: P = x y z Step 3: Identify the constraint: The constraint is given by: x + z = 12 and y + z = 12 Step 4: State the optimization problem: The given objective function is given by: Maximize the objective function P = x y z The constraint is given by: x + z = 12 and y + z = 12 , x , y , z ≥ 0 Step 5: Eliminate the extra variables, solve the constraint for one unknown and substitute in the objective function. The constraints are given by x + z = 12 and y + z = 12 Solving the first given constraint for x with respect to z to get: x = 12 − z Solving the second given constraint for y with respect to z to get: y = 12 − z Substituting 12 − z for x and 12 − z for y in the objective function to get: P = x y z P = ( 12 − z ) ( 12 − z ) z P = ( 12 − z ) 2 z Since x ≥ 0 Thus, 12 − z ≥ 0 12 ≥ z Thus, 0 ≤ z ≤ 12 Maximizing the objective function subject to 0 ≤ z ≤ 12 to get: P = ( 12 − z ) 2 z Differentiating both sides of the equation with respect to z to get: P ' ( z ) = ( 12 − z ) 2 − 2 z ( 12 − z ) P ' ( z ) = ( 12 − z ) ( 12 −

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.2, Problem 8E

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To calculate: The maximization of the optimization problem P=xyz subject to x+z=12 and y+z=12 when x,y,z≥0.

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Chapter 12 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

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The maximization of the optimization problem P = x y z subject to x + z = 12 and y + z = 12 when x , y , z ≥ 0 .

Solution Summary: The author explains the steps to solve optimization problems: Identify the unknowns, identify the objective function, solve the constraint, and find the absolute maximum or absolute minimum.

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To calculate: The maximization of the optimization problem P=xyz subject to x+z=12 and y+z=12 when x,y,z≥0.

Chapter 12 Solutions