To determine the concentration of the drug in the bloodstream after 2 minutes, we need to use the given formula for [tex]\( C(t) \)[/tex]. The formula is:
[tex]\[ C(t) = 0.07 \left(1 - e^{-0.2 t} \right) \][/tex]
Here's the step-by-step solution:
1. Substitute [tex]\( t = 2 \)[/tex] into the formula [tex]\( C(t) \)[/tex]:
[tex]\[ C(2) = 0.07 \left(1 - e^{-0.2 \cdot 2} \right) \][/tex]
2. Simplify the exponent:
[tex]\[ C(2) = 0.07 \left(1 - e^{-0.4} \right) \][/tex]
3. Calculate the value of the exponential term [tex]\( e^{-0.4} \)[/tex]:
Using a calculator or a software tool, we find:
[tex]\[ e^{-0.4} \approx 0.670320046 \][/tex]
4. Substitute the value of [tex]\( e^{-0.4} \)[/tex] back into the formula:
[tex]\[ C(2) = 0.07 \left(1 - 0.670320046 \right) \][/tex]
5. Subtract inside the parentheses:
[tex]\[ 1 - 0.670320046 \approx 0.329679954 \][/tex]
6. Multiply by 0.07:
[tex]\[ C(2) = 0.07 \times 0.329679954 \][/tex]
[tex]\[ C(2) \approx 0.02307759678 \][/tex]
7. Round the result to three decimal places:
[tex]\[ C(2) \approx 0.023 \][/tex]
Therefore, the concentration of the drug in the bloodstream after 2 minutes, rounded to three decimal places, is [tex]\( 0.023 \)[/tex].
This detailed step-by-step calculation shows that the concentration after 2 minutes is approximately [tex]\( \boxed{0.023} \)[/tex].
