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 Application Calculus (Looseleaf) - Text Only

7th Edition

ISBN: 9781337291484

Author: Waner

Publisher: Cengage

See similar textbooks

Videos

Question

Chapter 12.2, Problem 7E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.2, Problem 6E

Chapter 12.2, Problem 8E

Students have asked these similar questions

+ Find the first five non-zero terms of the Taylor series for f(x) = sin(2x) centered at 4π. + + + ...

+ + ... Find the first five non-zero terms of the Taylor series for f(x) centered at x = 4. = 1 x + + +

Find the interval and radius of convergence for the given power series. n=0 (− 1)" xn 7" (n² + 2) The series is convergent on the interval: The radius of convergence is R =

Chapter 12 Solutions

Finite Mathematics and Application Calculus (Looseleaf) - Text Only

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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

Solution Summary: The author calculates the maximization of the optimization problem P=xyz.

Videos

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

Want to see the full answer?

Chapter 12 Solutions