To determine the concentration [tex]\( C(t) \)[/tex] of the drug in the bloodstream after [tex]\( t = 3 \)[/tex] minutes, follow these steps:
1. Write down the given formula: The concentration [tex]\( C(t) \)[/tex] is given by the formula:
[tex]\[
C(t) = 0.08 \left(1 - e^{-0.2 t}\right)
\][/tex]
2. Substitute the given time into the formula: We need to find the concentration at [tex]\( t = 3 \)[/tex] minutes. Substitute [tex]\( t = 3 \)[/tex] into the formula:
[tex]\[
C(3) = 0.08 \left(1 - e^{-0.2 \cdot 3}\right)
\][/tex]
3. Calculate the exponent: Compute the exponent part inside the formula. First, calculate [tex]\( 0.2 \cdot 3 \)[/tex]:
[tex]\[
0.2 \times 3 = 0.6
\][/tex]
4. Evaluate the exponential function: Next, calculate [tex]\( e^{-0.6} \)[/tex]:
(We use a precise mathematical tool to get this value.)
5. Complete the computation inside the parentheses: Subtract the value of the exponential term from 1:
[tex]\[
1 - e^{-0.6}
\][/tex]
6. Multiply by the constant factor: Finally, multiply this result by 0.08 to get the concentration:
[tex]\[
C(3) = 0.08 \left(1 - e^{-0.6}\right)
\][/tex]
7. Round the result: Round the result to three decimal places.
After going through the calculation steps, the concentration of the drug in the bloodstream after 3 minutes is:
[tex]\[
C(3) = 0.036
\][/tex]
Thus, the concentration after 3 minutes is [tex]\(\boxed{0.036}\)[/tex].
