Question

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. s(t)=15 - 13e^{-0.1t} 0≤t≤20 (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? s(t)=15t + 130e^{-0.1t} - 130 (b) what are the estimated total sales for the first 10 months of the campaign? □ million (round to the nearest whole number.)

Explanation:

Step1: Substitute \( t = 10 \) into \( S(t) \)

Given \( S(t)=15t + 130e^{-0.1t}-130 \), when \( t = 10 \), we have \( S(10)=15\times10+130e^{-0.1\times10}-130 \).
First, calculate \( 15\times10 = 150 \) and \( - 0.1\times10=-1 \), so \( S(10)=150+130e^{-1}-130 \).
Simplify \( 150 - 130=20 \), then \( S(10)=20 + 130e^{-1} \).
Since \( e^{-1}=\frac{1}{e}\approx0.3679 \), then \( 130e^{-1}=130\times0.3679 = 47.827 \).
So \( S(10)=20 + 47.827=67.827 \).

Step2: Round the result

Rounding \( 67.827 \) to the nearest whole number.
Since the decimal part \( 0.827>0.5 \), we round up.

Answer:

\( 68 \) million.