Calculate the price, advertising levels that maximize sales

[gary_]

A firm faces the following sales function:

. . .Q = 5,000 - 10P + 40A + PA - 0.8A^2 - 0.5P^2

...where Q is sales, P is the price (in dollars), and A is the advertising expenditures (in hundreds of dollars). Calculate the price and advertising levels that maximize sales.

I don't know what to solve it with three variables.

 

[soroban]

Re: Calculate the price, advertising levels that maximize sa

Hello, Gary!

If you are not familiar with partial derivatives,
. . you shouldn't be assigned this problem.

A firm faces the following sales function:
. . .\(\displaystyle Q \:= \:5,000\,-\,10P\,+\,40A\,+\,PA\,-\,0.8A^2\,-\,0.5P^2\)
where \(\displaystyle Q\) is sales, \(\displaystyle P\) is the price (in dollars),
and \(\displaystyle A\) is the advertising expenditures (in hundreds of dollars).
Calculate the price and advertising levels that maximize sales.

Find the two partial derivatives and equate to zero.

\(\displaystyle \begin{array}{cc}\frac{\partial Q}{\partial P}& \.=\. & -10\,+\,A\,-\,P & \.=\. & 0 &\;\;\Rightarrow\;\;& A\,-\,P & \,=\, & 10 & \;(1)\\
\frac{\partial Q}{\partial A} & \,=\, & 40\,+\,P\,-\,1.6A & \,=\, & 0 & \;\;\Rightarrow\;\;& -1.6A\,+\,P & \,=\, & -40 & \;(2)
\end{array}\)


Add (1) and (2): \(\displaystyle \:-0.6A\:=\:-30\;\;\Rightarrow\;\;\fbox{A\,=\,50}\)

Substitute into (1): \(\displaystyle \:50\,-\,P\:=\:10\;\;\Rightarrow\;\;\fbox{P\,=\,40}\)

 

[gary_]

Its just an extra problem for us.
Thanks.