To determine the concentration [tex]\( C(t) \)[/tex] of the drug in the bloodstream after [tex]\( t = 20 \)[/tex] minutes, we use the provided formula:
[tex]\[ C(t) = 0.08 \left(1 - e^{-0.2t}\right) \][/tex]
Step-by-Step Solution:
1. Substitute the given time [tex]\( t = 20 \)[/tex] minutes into the formula:
[tex]\[ C(20) = 0.08 \left(1 - e^{-0.2 \times 20}\right) \][/tex]
2. Simplify the exponent calculation inside the exponentiation function:
[tex]\[ C(20) = 0.08 \left(1 - e^{-4}\right) \][/tex]
3. Calculate the exponential term [tex]\( e^{-4} \)[/tex]:
[tex]\[ e^{-4} \approx 0.0183 \][/tex]
4. Subtract this result from 1:
[tex]\[ 1 - 0.0183 = 0.9817 \][/tex]
5. Multiply the result by 0.08 to find the concentration:
[tex]\[ C(20) = 0.08 \times 0.9817 \approx 0.0785 \][/tex]
6. Round the concentration to three decimal places:
[tex]\[ C(20) \approx 0.079 \][/tex]
Therefore, the concentration of the drug in the bloodstream after 20 minutes is approximately [tex]\( 0.079 \)[/tex].
Answer:
[tex]\[ \text{drug concentration} = 0.079 \][/tex]
