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 $f (t) = 100 (1 - e^{- 0.05 t})$ , where $t$ 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

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Final Answer: $32$ days.

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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 $f (t) = 100 (1 - e^{- 0.05 t})$ .

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: $32$ days.

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