GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $2,500 Is available for this promotional campaign. Based on pas experience in promoting Its other services, GreenLawns obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media. S = -2R²10M2 - 8RM+ 18R+ 34M where S = total sales in thousands of dollars R= thousands of dollars spent on radio advertising M= thousands of dollars spent on direct-mail advertising GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget. (a) What is the value of sales (in dollars) if $1,500 is spent on radio advertising and $1,000 is spent on direct-mall advertising? S = $ X (b) Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $2,500 on total advertising. Max s.t. R, M20 X x ≤ 2.50 (c) Determine the optimal amount to spend (in dollars) on radio and direct-mail advertising. (Round your answers to the nearest dollar.)
GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $2,500 Is available for this promotional campaign. Based on pas experience in promoting Its other services, GreenLawns obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media. S = -2R²10M2 - 8RM+ 18R+ 34M where S = total sales in thousands of dollars R= thousands of dollars spent on radio advertising M= thousands of dollars spent on direct-mail advertising GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget. (a) What is the value of sales (in dollars) if $1,500 is spent on radio advertising and $1,000 is spent on direct-mall advertising? S = $ X (b) Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $2,500 on total advertising. Max s.t. R, M20 X x ≤ 2.50 (c) Determine the optimal amount to spend (in dollars) on radio and direct-mail advertising. (Round your answers to the nearest dollar.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**GreenLawns Promotional Strategy**
GreenLawns provides lawn fertilizing and weed control services. The company is introducing a special aeration treatment as a low-cost extra service, expected to attract new customers. They plan to promote this new service through radio and direct-mail advertising, with a media budget of $2,500. Based on past advertising experiences, GreenLawns has estimated the relationship between sales and promotional spending in these media:
\[ S = -2R^2 - 10M^2 - 8RM + 18R + 34M \]
Where:
- \( S \) = total sales in thousands of dollars
- \( R \) = thousands of dollars spent on radio advertising
- \( M \) = thousands of dollars spent on direct-mail advertising
**Objectives:**
GreenLawns aims to develop a promotional strategy that maximizes sales within the given media budget constraints.
**Tasks:**
(a) Calculate sales when $1,500 is spent on radio advertising and $1,000 on direct-mail advertising.
*S = $* \([ \_\_\_\_\_\_\ ]\)
(b) Formulate an optimization problem to maximize sales, ensuring the total advertising spend does not exceed $2,500.
\[ \text{Max} \]
\[ \text{s.t. } \]
\[ M \]
\[ \leq 2.50 \]
\[ R, M \geq 0 \]
(c) Determine the optimal amounts to spend (in dollars) on radio and direct-mail advertising. Round your answers to the nearest dollar.
- **Radio:** \( R = \$* \([ \_\_\_\_\_\_\ ]\)
- **Direct-Mail:** \( M = \$* \([ \_\_\_\_\_\_\ ]\)
**Calculate Total Sales:** How much money (in dollars) will be generated in sales? Round your answer to the nearest dollar.
- **Total Sales:** \( S = \$* \([ \_\_\_\_\_\_\ ]\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15c8605-c3c4-4922-bcea-8a06f4b6dc06%2F5fb37f3b-346d-4ed1-9c96-e0054dddbe43%2Fm2ip96_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**GreenLawns Promotional Strategy**
GreenLawns provides lawn fertilizing and weed control services. The company is introducing a special aeration treatment as a low-cost extra service, expected to attract new customers. They plan to promote this new service through radio and direct-mail advertising, with a media budget of $2,500. Based on past advertising experiences, GreenLawns has estimated the relationship between sales and promotional spending in these media:
\[ S = -2R^2 - 10M^2 - 8RM + 18R + 34M \]
Where:
- \( S \) = total sales in thousands of dollars
- \( R \) = thousands of dollars spent on radio advertising
- \( M \) = thousands of dollars spent on direct-mail advertising
**Objectives:**
GreenLawns aims to develop a promotional strategy that maximizes sales within the given media budget constraints.
**Tasks:**
(a) Calculate sales when $1,500 is spent on radio advertising and $1,000 on direct-mail advertising.
*S = $* \([ \_\_\_\_\_\_\ ]\)
(b) Formulate an optimization problem to maximize sales, ensuring the total advertising spend does not exceed $2,500.
\[ \text{Max} \]
\[ \text{s.t. } \]
\[ M \]
\[ \leq 2.50 \]
\[ R, M \geq 0 \]
(c) Determine the optimal amounts to spend (in dollars) on radio and direct-mail advertising. Round your answers to the nearest dollar.
- **Radio:** \( R = \$* \([ \_\_\_\_\_\_\ ]\)
- **Direct-Mail:** \( M = \$* \([ \_\_\_\_\_\_\ ]\)
**Calculate Total Sales:** How much money (in dollars) will be generated in sales? Round your answer to the nearest dollar.
- **Total Sales:** \( S = \$* \([ \_\_\_\_\_\_\ ]\)
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