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 $2.8 million decline in sales of brand Z, whereas each $1 million increase in sales of brand Y results in an increase of $0.5 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).)

Answer 1

Let Z = Z₀ + a(X-X₀) + b(Y-Y₀), where X, Y and Z are sales in million dollars.

Where 'Z₀' is a constant which is starting point of 'Z'.

'a' is the rate of change of Z with respect to 'X'

'b' is the rate of change of Z with respect to 'Y'

and, X₀ and Y₀ are the starting point of 'X' and 'Y' respectively.

We are given that each $1 million increase in sales of brand X causes a $2.8 million decline in sales of brand Z, whereas each $1 million increase in sales of brand Y results in an increase of $0.5 million in sales of brand Z.

So, we can say that -

a = -2.8 and b = 0.5

Also, all the three variables are starting at 6.

So, X₀ = 6, Y₀ = 6 and Z₀ = 6.

Thus, the equation is -

Z = 6 - 2.8(X - 6) + 0.5(Y - 6)

= 6 - 2.8(X) + 16.8 + 0.5(Y) - 3

So, Z = 19.8 - 2.8(X) + 0.5(Y) million dollars.