GreenLawns provides a lawn fertilizer 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 $3,500 is available for this promotional campaign. Based on past 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 = −2R2 − 10M2 − 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 $2,500 is spent on radio advertising and $1,000 is spent on direct-mail advertising? S = $ (b) Formulate an optimization problem that can be solved to maximize sales subject to the media budget. Max −2R2−10M2−8RM+18R+34M s.t. R+M ≤ 3.5 (c) Determine the optimal amount to spend (in dollars) on radio and direct-mail advertising. (Round your answers to the nearest dollar.) radio R =$ direct-mail M =$ How much money (in dollars) will be generated in sales? Total sales

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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Question

GreenLawns provides a lawn fertilizer 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 $3,500 is available for this promotional campaign. Based on past 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 = −2R² − 10M² − 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 $2,500 is spent on radio advertising and $1,000 is spent on direct-mail advertising?

S = $

(b)

Formulate an optimization problem that can be solved to maximize sales subject to the media budget.