Brand Z\'s annual sales are affected by the sales of related products X and Y as

Question

Brand Z's annual sales are affected by the sales of related products X and Y as follows: Each $1 million increase in sales of brand X causes a $2.5 million decline in sales of brand Z, whereas each $1 million increase in sales of brand Y results in an increase of $0.9 million in sales of brand Z. Currently, brands X, Y, and Z are each selling $6 million per year. Model the sales of brand Z using a linear function. (Let z = annual sales of Z (in millions of dollars), x = annual sales of X (in millions of dollars), and y = annual sales of Y (in millions of dollars).)

z=??

Explanation / Answer

The most general form for z as a linear function of x and y is

z = ax + by + c

I assume you know about partial derivatives. Note that ?z/?x = a. This is a measure of how much z changes per unit change in x. We're told that z declines 2.5 when x has a unit increase, so

a = -2.5

Similarly, since z increases by 0.9 for a unit increase in y, we have

b = +0.9

so

z = -2.5x + 0.9y + c.

We are also told that z = 6 when x = y = 6. That enables us to determine c. You should get c = 15.6

Note that this could be written in the slightly more intuitive form

(z - 6) = -2.5(x - 6) + 0.9(y - 6)

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