Explanation Given information: A firm markets a product in two countries. Let x be the number of units to be sold in the first country and y be the number of unit to be sold in the second country. The firm must set the same price in each country, that is, 97 − ( x 10 ) = 83 − ( y 20 ) dollars. The cost of producing these units is 20 , 000 + 3 ( x + y ) Explanation: The objective function is a profit function and is found by the formula, Profit = Revenue - Cost f ( x , y ) = ( 97 − ( x 10 ) ) x + ( 83 − ( y 20 ) ) y − ( 20000 + 3 ( x + y ) ) ⇒ 97 x − x 2 10 + 83 y − y 2 20 − 20 , 000 − 3 x − 3 y ⇒ 94 x − x 2 10 + 80 y − y 2 20 − 20 , 000 And the condition is that the firm must set the same price in each country, that is, 97 − ( x 10 ) = 83 − ( y 20 ) dollars. So, the constraint equation is g ( x , y ) = 97 − 83 − x 10 + y 20 = 0 ⇒ g ( x , y ) = 14 − x 10 + y 20 = 0 Therefore, the objective function is f ( x , y ) = 94 x − x 2 10 + 80 y − y 2 20 − 20 , 000 subject to the constraint g ( x , y ) = 14 − x 10 + y 20 = 0 Use Lagrange multipliers and constrained optimization method: Replace f ( x , y ) by an auxiliary function of three variables F ( x , y , λ ) as, F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Substitute f ( x , y ) = 94 x − x 2 10 + 80 y − y 2 20 − 20 , 000 and g ( x , y ) = 14 − x 10 + y 20 = 0 in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y )

12th Edition

ISBN: 9780137590810

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

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 7.4, Problem 26E

To determine

The value of x,y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97−(x10)=83−(y20).

The cost of producing these units is 20,000+3(x+y)

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.4, Problem 25E

Chapter 7.4, Problem 27E

Students have asked these similar questions

5:38 Video Message instructor Submit Question ||| D

8:38 *** TEMU TEMU -3 -2 7 B 2 1 & 5G. 61% 1 2 -1 Based on the graph above, determine the amplitude, period, midline, and equation of the function. Use f(x) as the output. Amplitude: 2 Period: 2 Midline: 2 ☑ syntax error: this is not an equation. Function: f(x) = −2 cos(πx + 2.5π) +2× Question Help: Worked Example 1 ☑ Message instructor Submit Question ||| <

8:39 *** TEMU 5G 60% A ferris wheel is 28 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the amplitude? 14 meters What is the equation of the Midline? y = 16 What is the period? 4 meters minutes The equation that models the height of the ferris wheel after t minutes is: f(t): = ƒ (3) = ·−14(0) + 16 syntax error: you gave an equation, not an expression. syntax error. Check your variables - you might be using an incorrect one. How high are you off of the ground after 3 minutes? Round your answe the nearest meter. ||| <

Chapter 7 Solutions

Calculus & Its Applications

Ch. 7.1 - Let f(x,y,z)=x2+y(xz)4. Compute f(3,5,2).Ch. 7.1 - Prob. 2CYU Ch. 7.1 - Let f(x,y)=x23xyy2. Compute f(5,0),f(5,2) and...Ch. 7.1 - Prob. 2E Ch. 7.1 - Let g(x,y,z)=x/(yz). Compute g(1,1),g(0,1) and...Ch. 7.1 - Let f(x,y,z)=x2ey2+z2. Compute f(1,1,1) and...Ch. 7.1 - Let f(x,y)=xy. Show that f(2+h,3)f(2,3)=3h.Ch. 7.1 - Let f(x,y)=xy. Show that f(2,3+k)f(2,3)=2k.Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...Ch. 7.1 - Cost Find a formula C(x,y,z) that gives the cost...

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 value of x , y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97 − ( x 10 ) = 83 − ( y 20 ) . The cost of producing these units is 20 , 000 + 3 ( x + y )

Solution Summary: The author explains that the objective function is a profit function. The firm must set the same price in each country.

Concept explainers

Videos

The value of x,y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97−(x10)=83−(y20).

The cost of producing these units is 20,000+3(x+y)

Want to see the full answer?

Chapter 7 Solutions

The value of x , y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97 − ( x 10 ) = 83 − ( y 20 ) . The cost of producing these units is 20 , 000 + 3 ( x + y )

Solution Summary: The author explains that the objective function is a profit function. The firm must set the same price in each country.

Concept explainers

Videos

The value of x,y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97−(x10)=83−(y20). The cost of producing these units is 20,000+3(x+y)

Want to see the full answer?

Chapter 7 Solutions

The value of x,y that maximizes the profit, where a firm markets a product in two countries and set the same price in each country using the equation 97−(x10)=83−(y20).

The cost of producing these units is 20,000+3(x+y)