Bounds Inc.has determined through regression analysis that its sales (S) are a f

Question

Bounds Inc.has determined through regression analysis that its sales (S) are a
function ofthe amount ofadvertising (measured in units) in two different media.
This is given by the following relationship (X=newspapers,Y=magazines):
S(X,Y) = 200X + 100Y - 10X^2 - 20Y^2 + 20XY
a. Find the level ofnewspaper and magazine advertising that maximizes the
firm’s sales.
I know how to do this part - I found the partial derivatives of X and Y. Need help with part B.

b. Calculate the firm’s sales at the optimal values ofnewspaper and magazine
advertising determined in part (a).

The answers log shows that we plug in values Y = 15 units and X = 25 units. What I don't understand is where these values come from?

Explanation / Answer

A. I'm just going to check your work. S = 200X + 100Y - 10X^2 - 20Y^2 + 20XY The partial derivatives are calculated and set to zero: Sx = 200 - 20X + 20Y = 0 (1) Sy = 100 - 40Y + 20X = 0 (2) Now, we can solve for the X and Y that maximize sales. From (1), solve for X 200 - 20X + 20Y = 0 20X = 200 + 20Y X = 10 + Y (3) Substitute into (2) and solve for Y 100 - 40Y + 20X = 0 100 - 40Y + 20(10 + Y) = 0 100 - 40Y + 200 + 20Y = 0 300 = 20Y Y = 300/20 Y = 15 Substitute this into (3) to solve for X X = 10 + Y X = 10 + 15 X = 25 This means that X = 25 and Y = 15 maximize sales. B. Here, we just plug these values into sales to see what the maximum sales are. S = 200X + 100Y - 10X^2 - 20Y^2 + 20XY S = 200*25 + 100*15 - 10*(25^2) - 20*(15^2) + 20*25*15 S = 3,250

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