(a) Explanation Given: The Fancy Fashions, the local boutique, has $ 8000 available each month for advertisement. Newspaper ads cost $ 400 each, and no more than 30 can run per month. Internet banner ads cost $ 20 each, and no more than 60 can run per month. TV ads cost $ 2000 each, with a maximum of 10 available per month. Approximately, 4000 women see each newspaper ad, 3000 see each internet banner, and 10,000 see each TV advertisement. Explanation: Consider The number of newspaper advertisement is equal to x 1 . The number of internet banner advertisement is equal to x 2 . The number of TV advertisement is equal to x 3 . The constraints from problemare as follows: 400 x 1 + 20 x 2 + 2000 x 3 ≤ 8000 x 1 ≤ 30 x 2 ≤ 60 x 3 ≤ 10 The objective function is z = 4000 x 1 + 3000 x 2 + 10000 x 3 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Now, apply the simplex method for maximization. Step 1: Add slack variables or basic variable to each constraint to convert it into linear equation. Basic variables are the variables that are nonzero. As a result, 400 x 1 + 20 x 2 + 2000 x 3 + s 1 = 8000 x 1 + s 2 = 30 x 2 + s 3 = 60 x 3 + s 4 = 10 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Here s 1 , s 2 , s 3 , a n d s 4 are basic variables greater than or equal to zero and x 1 , x 2 , x 3 , and x 4 are nonbasic variables. Step 2: Represent the system of linear equations obtained above in simplex tableau. This is initial tableau. [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 z s 1 400 20 2000 1 0 0 0 0 8000 s 2 1 0 0 0 1 0 0 0 30 s 3 0 1 0 0 0 1 0 0 60 s 4 0 0 1 0 0 0 1 0 10 z − 4000 − 3000 − 10000 0 0 0 0 1 0 ] _ This tableau leads to the solution x 1 , x 2 , x 3 ≥ 0 and s 1 = 8000 , s 2 = 30 , s 3 = 60 , s 4 = 10 with z = 0 Step 3: Select a pivot column. Select the negative entry in the bottom row with the largest numerical value. In initial tableau, the negative entry in the bottom row with the largest numerical value is − 10 , 000 .Hence, the column corresponding to − 10 , 000 is the pivot column, which follows that x 3 is enter variable. Step 4: Select the pivot. Since the pivot must be a positive number, apply test ratio for the pivot. 8000 2000 = 4 , 10 1 = 10 Therefore, 2000 is pivot because the smallest positive ratio is obtained by the element of R 1 , which follows that s 1 is a depart variable. Step 5: Use the pivot to make other entries in the pivot column zero. For this, first, if pivot element is not 1 , make pivot element 1 by R 1 → R 1 2000 : [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 z x 3 1 5 1 100 1 1 2000 0 0 0 0 4 s 2 1 0 0 0 1 0 0 0 30 s 3 0 1 0 0 0 1 0 0 60 s 4 0 0 1 0 0 0 1 0 10 z − 4000 − 3000 − 10 , 000 0 0 0 0 1 0 ] _ Now make other entries in the pivot column zero. R 4 → R 4 − R 1 R 5 → R 5 + 10 , 000 R 1 This leads to [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 z x 3 1 5 1 100 1 1 2000 0 0 0 0 4 s 2 1 0 0 0 1 0 0 0 30 s 3 0 1 0 0 0 1 0 0 60 s 4 − 1 5 − 1 100 0 − 1 2000 0 0 1 0 6 z − 2000 − 2900 0 5 0 0 0 1 40 , 000 ] _ Again, choose pivot column. Column 2 is the pivot. Pivot number is 4 1 100 = 400 , 60 1 = 60 Thus, 1 is pivot; that is, x 2 is enter variable and s 3 a depart variable (b) To determine Each type of advertisement required to obtain the maximum profit.

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 4.2, Problem 29E

(a)

To determine

Each type of advertisement required to obtain the maximum profit.

(b)

To determine

Each type of advertisement required to obtain the maximum profit.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 28E

Chapter 4.2, Problem 30E

Students have asked these similar questions

2. Let Y₁,……., Y be a random sample with common mean μ and common variance σ². Use the CLT to write an expression approximating the CDF P(Ỹ ≤ x) in terms of µ, σ² and n, and the standard normal CDF Fz(·).

3. We'd like to know the first time when the population reaches 7000 people. First, graph the function from part (a) on your calculator or Desmos. In the same window, graph the line y = 7000. Notice that you will need to adjust your window so that you can see values as big as 7000! Investigate the intersection of the two graphs. (This video shows you how to find the intersection on your calculator, or in Desmos just hover the cursor over the point.) At what value t> 0 does the line intersect with your exponential function? Round your answer to two decimal places. (You don't need to show work for this part.) (2 points)

Suppose the planet of Tattooine currently has a population of 6500 people and an annual growth rate of 0.35%. Use this information for all the problems below. 1. Find an exponential function f(t) that gives the population of Tattooine t years from now. (3 points)

Chapter 4 Solutions

Finite Mathematics (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

Knowledge Booster

Learn more about

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

Each type of advertisement required to obtain the maximum profit.

Solution Summary: The author explains how each type of advertisement required to obtain maximum profit. They add slack variables or basic variables to each constraint to convert it into linear equation.

Videos

Each type of advertisement required to obtain the maximum profit.

Each type of advertisement required to obtain the maximum profit.

Want to see the full answer?

Chapter 4 Solutions