Question

A manufacturing company is ready to introduce a new product with a national sales campaign. After extensive test marketing, the market research department estimates that sales (in millions of dollars) will increase at the monthly rate of S'(t) = 10 - 10e-0.2t for 0 ≤ t ≤ 24, t months after the national campaign has started. What will the total sales be five months after the beginning of the campaign if we assume zero sales at the beginning of the campaign? (Round the answer to the nearest million.)

Answer 1

S(t) = 68 million dollars

the total sales five months after the beginning of the campaign is 68 million dollars

Explanation:

Given that sales (in millions of dollars) will increase at the monthly rate of S'(t) = 10 - 10e^-0.2t for 0 ≤ t ≤ 24, t months after the national campaign has started

Change in sales is;

S'(t) = 10 - 10e^-0.2t

The total sales at time t from the beginning of the sales campaign is;

S(t) = ∫S'(t) = ∫(10 - 10e^-0.2t)

S(t) = 10t + (10/0.2)e^-0.2t + S₀

S(t) = 10t + 50e^-0.2t + S₀

Since we assume zero sales at the beginning of the campaign

S₀ = 0

S(t) = 10t + 50e^-0.2t

Given;

Time t = 5 months

Substituting the values into the equation;

S(t) = 10(5) + 50e^-0.2(5)

S(t) = 68.39

S(t) = 68 million dollars

the total sales five months after the beginning of the campaign is 68 million dollars