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 $$6,000$ is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S $=-2$R$212$M$29$RM $+16$R $+34$M$,$

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 if $$5,000$ is spent on radio advertising and $$1,000$ is spent on direct$-$mail advertising? Enter amounts in dollars, i$.$ e$.$ $$12$ thousands should be written as $$12,000$

$

$($b$)$ Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $$6,000$ on total advertising. If the constant is $"1"$ it must be entered in the box. For subtractive or negative numbers use a minus sign even if there is a $+$ sign before the blank $($Example: $-300).$ If your answer is zero enter $0.$

Max

$-2$

R $2+$

$-12$

M $2+$

$-9$

RM $+$

$16$

R $+$

$34$

M

s$.$t$.$

$1$

R $+$

$1$

M

$<=$

$6$

Budget

R$,$ M

$>=$

$0$

$($c$)$ Determine the optimal amount to spend on radio and direct$-$mail advertising. How much in sales will be generated? Enter amounts in dollars, i$.$ e$.$ $$12$ thousands should be written as $$12,000.$ If your answer is zero enter $0.$

Amount spent on radio advertising $=$ $

$4000$

Amount spent on direct$-$mail advertising $=$ $

$0$

Total Sales $=$ $