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.

Question

Alicia Reyes

Mathematics

21 October, 04:44

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.

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Dennis Dougherty · Answer 1

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)