To find the concentration [tex]\( C(t) \)[/tex] of the drug in the bloodstream after 9 minutes using the given formula, [tex]\( C(t) = 0.05 \left( 1 - e^{-0.2t} \right) \)[/tex], follow these steps:
1. Substitute the value of [tex]\( t \)[/tex] into the formula:
The given time is [tex]\( t = 9 \)[/tex] minutes. Substitute [tex]\( t = 9 \)[/tex] into the equation:
[tex]\[
C(9) = 0.05 \left( 1 - e^{-0.2 \cdot 9} \right)
\][/tex]
2. Calculate the exponent:
Calculate [tex]\( -0.2 \cdot 9 \)[/tex]:
[tex]\[
-0.2 \cdot 9 = -1.8
\][/tex]
3. Evaluate the exponential function:
Calculate [tex]\( e^{-1.8} \)[/tex]:
[tex]\[
e^{-1.8} \approx 0.16529888822158656
\][/tex]
4. Subtract the result from 1:
Substitute the value of [tex]\( e^{-1.8} \)[/tex] back into the equation:
[tex]\[
1 - e^{-1.8} = 1 - 0.16529888822158656 \approx 0.8347011117784134
\][/tex]
5. Multiply by 0.05:
Finally, multiply the result by 0.05:
[tex]\[
0.05 \cdot 0.8347011117784134 \approx 0.041735055588920676
\][/tex]
6. Round to three decimal places:
Now, round the concentration to three decimal places:
[tex]\[
0.041735055588920676 \approx 0.042
\][/tex]
Therefore, the concentration [tex]\( C(9) \)[/tex] of the drug in the bloodstream after 9 minutes is approximately [tex]\( \boxed{0.042} \)[/tex] (rounded to three decimal places).
