Total sales of a fad product S may increase slowly at first while the product is relatively unknown, more rapidly as its popularity rises, and then level of toward a ;og term maximum as consumers mive onto another product. Consider the model In(1-S)-In(S) kt +C, where S is the fraction of total sales (S 1 means 100%), t weeks after the product is introduced, and k and C are constants. Solve the model for sales S in terms of k, t, and C.

Sales, S(t)= 1/(Ceᵏᵗ+1)

Step-by-step explanation:

In(1-S)-In(S) = kt +C

Applying Laws of Logarithm

ln A- ln B= ln (A/B)

In(1-S)-In(S) = kt +C

ln ((1-S)/S) =kt + C

Taking the exponential of both sides

(1-S)/S = eᵏᵗ⁺ᶜ

(1-S)/S = eᵏᵗ*eᶜ

Now, exponential of a constant,

eᶜ = C where C is an arbitrary constant.

(1-S)/S = Ceᵏᵗ

1-S= SCeᵏᵗ

SCeᵏᵗ +S=1

S(Ceᵏᵗ+1)=1

S(t)= 1/(Ceᵏᵗ+1)

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