A company begins a radio advertising campaign in Chicago to market a new soft drink. The percentage of the target market that buys a soft drink is estimated by the function $\mathrm{f}(\mathrm{t})=100\left(1-e^{-0.05 t}\right)$, where $\mathrm{t}$ is the number of days of the campaign. After how long will $80 \%$ of the target market have bought the soft drink?
A. 32 days
B. 80 days
C. 3 days
D. 33 days
Answer
Final Answer: \(\boxed{32}\) days.
Step 1 ：The problem is asking for the time it takes for 80% of the target market to have bought the soft drink. This means we need to solve the equation \(f(t) = 80\) for \(t\). The equation is given by \(f(t) = 100(1 - e^{-0.05t})\).
Step 2 ：We can rearrange this equation to solve for \(t\).
Step 3 ：\(t = -\frac{\ln(1-\frac{80}{100})}{0.05}\)
Step 4 ：By calculating the above expression, we get \(t\approx 32.1887582486820\)
Step 5 ：Rounding to the nearest whole number, we get \(t = 32\) days.
Step 6 ：Final Answer: \(\boxed{32}\) days.
