To solve this problem, we need to evaluate the given function [tex]\( f(t) = 100 \left( 1 - e^{-0.05 t} \right) \)[/tex] at [tex]\( t = 10 \)[/tex] days. Let's go through the steps:
1. Substitute [tex]\( t = 10 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
[tex]\[
f(10) = 100 \left( 1 - e^{-0.05 \cdot 10} \right)
\][/tex]
2. Simplify the exponent:
[tex]\[
0.05 \times 10 = 0.5
\][/tex]
So, the function [tex]\( f(10) \)[/tex] becomes:
[tex]\[
f(10) = 100 \left( 1 - e^{-0.5} \right)
\][/tex]
3. Evaluate the exponential part [tex]\( e^{-0.5} \)[/tex]:
Upon calculation, [tex]\( e^{-0.5} \approx 0.6065306597126334 \)[/tex].
4. Subtract the result from 1:
[tex]\[
1 - 0.6065306597126334 = 0.3934693402873666
\][/tex]
5. Multiply by 100 to get the percentage:
[tex]\[
100 \times 0.3934693402873666 \approx 39.346934028736655
\][/tex]
6. Round to the nearest hundredth:
[tex]\[
39.346934028736655 \approx 39.35
\][/tex]
So, the percentage of the target market that is predicted to buy the game after a 10-day campaign is 39.35%.
