an automobile company is ready to introduce a new line of cars through a national sales campaign. after test…

an automobile company is ready to introduce a new line of cars through a national sales campaign. after test marketing the line in a carefully selected city, the marketing research department estimates that the sales (in millions of dollars) will increase at the monthly rate of the following function for t months after the campaign has started.\n\ns(t)=15 - 13e^{-0.1t}\n\n0 ≤ t ≤ 20\n\n(a) what will be the total sales, s(t), t months after the beginning of the national sales campaign if we assume no sales at the beginning of the campaign?\n\ns(t)=□

an automobile company is ready to introduce a new line of cars through a national sales campaign. after test marketing the line in a carefully selected city, the marketing research department estimates that the sales (in millions of dollars) will increase at the monthly rate of the following function for t months after the campaign has started.\n\ns(t)=15 - 13e^{-0.1t}\n\n0 ≤ t ≤ 20\n\n(a) what will be the total sales, s(t), t months after the beginning of the national sales campaign if we assume no sales at the beginning of the campaign?\n\ns(t)=□

Answer

Explanation:

Step1: Integrate the rate function

We know that (S(t)=\int S^{\prime}(t)dt). Given (S^{\prime}(t)=15 - 13e^{-0.1t}), then (S(t)=\int(15 - 13e^{-0.1t})dt). Using the integral rules (\int a dt=at + C) ((a = 15)) and (\int e^{kt}dt=\frac{1}{k}e^{kt}+C) ((k=- 0.1)), we have: (S(t)=\int15dt-\int13e^{-0.1t}dt) (S(t)=15t-13\times\frac{e^{-0.1t}}{-0.1}+C) (S(t)=15t + 130e^{-0.1t}+C)

Step2: Determine the constant (C)

Since there are no sales at the beginning ((t = 0), (S(0)=0)). Substitute (t = 0) and (S(0)=0) into (S(t)=15t + 130e^{-0.1t}+C): (0=15\times0+130e^{0}+C) Since (e^{0}=1), we get (0 = 130 + C), so (C=-130)

Step3: Write the final function

Substitute (C = - 130) into (S(t)=15t + 130e^{-0.1t}+C) (S(t)=15t+130e^{-0.1t}-130)

Answer:

(S(t)=15t + 130e^{-0.1t}-130)