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 + y = 30 , y + z = 30 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 + y = 30 , y + z = 30 , 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 + y = 30 , y + z = 30 Solving the first given constraint for x with respect to y to get: x = 30 − y Solving the second given constraint for z with respect to y to get: z = 30 − y Substituting 30 − y for x and 30 − y for z in the objective function to get: P = x y z P = ( 30 − y ) y ( 30 − y ) P = ( 30 − y ) 2 y Since x ≥ 0 Thus, 30 − y ≥ 0 30 ≥ y Thus, 0 ≤ y ≤ 30 Maximizing the objective function subject to 0 ≤ y ≤ 30 to get: P = ( 30 − y ) 2 y Differentiating both sides of the equation with respect to y to get: P ' ( y ) = ( 30 − y ) 2 − 2 y ( 30 − y ) P ' ( y ) = ( 30 − y ) ( 30 − y

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 7E

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To calculate: The maximization of the optimization problem P=xyz subject to x+y=30 and y+z=30 when x,y,z≥0.

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Chapter 12.2, Problem 6E

Chapter 12.2, Problem 8E

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The maximization of the optimization problem P = x y z subject to x + y = 30 and y + z = 30 when x , y , z ≥ 0 .

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To calculate: The maximization of the optimization problem P=xyz subject to x+y=30 and y+z=30 when x,y,z≥0.

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Chapter 12 Solutions