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 $2.6 million decline in sales of brand Z, whereas each $1 million increase in sales of brand Y results in an increase of $0.3 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 general form for z as a linear function of x and y is z=ax+by+c

partial derivating with respect to x and y

dz/dx=a=-2.6 since this is a measure how much z changes per unit change in x.We're told that z declines 2.6 when x has a unit increase

dz/dy=b=0.3 since this is a measure how much z changes per unit change in y.We're told that z increases 0.3 when y has a unit increase

substitute a,b into z

z=-2.6x+0.3y+c --(1)

we are also told that z=6 when x=6,y=6

substitute z=6,x=6 and y=6 into equation (1)

6=-2.6*6+0.3*6+c

6=-15.6+1.8+c

c=6+15.6-1.8

c=19.8

substitute x=19.8 into equation 1

z=-2.6x+0.3y+19.8

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