To find the concentration [tex]\( C(t) \)[/tex] of the drug in the bloodstream after [tex]\( t \)[/tex] minutes, we use the given formula:
[tex]\[ C(t) = 0.04 \left( 1 - e^{-0.2t} \right) \][/tex]
We need to determine the concentration when [tex]\( t = 5 \)[/tex] minutes. Follow these steps to solve it:
1. Substitute [tex]\( t \)[/tex] with 5: Replace [tex]\( t \)[/tex] in the formula with 5.
[tex]\[
C(5) = 0.04 \left( 1 - e^{-0.2 \cdot 5} \right)
\][/tex]
2. Simplify the exponent: Calculate the exponent [tex]\( -0.2 \cdot 5 \)[/tex].
[tex]\[
-0.2 \cdot 5 = -1
\][/tex]
So, the expression inside the exponential function becomes [tex]\( e^{-1} \)[/tex].
3. Calculate the exponential: Determine the value of [tex]\( e^{-1} \)[/tex]. The value of [tex]\( e \)[/tex] (Euler's number) is approximately 2.71828, and [tex]\( e^{-1} \)[/tex] is approximately [tex]\( 1 / 2.71828 \)[/tex].
[tex]\[
e^{-1} \approx 0.367879
\][/tex]
4. Substitute [tex]\( e^{-1} \)[/tex]: Replace [tex]\( e^{-1} \)[/tex] with its approximate value in the expression.
[tex]\[
C(5) = 0.04 \left( 1 - 0.367879 \right)
\][/tex]
5. Perform the subtraction: Subtract the approximate value of [tex]\( e^{-1} \)[/tex] from 1.
[tex]\[
1 - 0.367879 = 0.632121
\][/tex]
6. Multiply by 0.04: Multiply the result by 0.04 to find the concentration.
[tex]\[
C(5) = 0.04 \times 0.632121 \approx 0.0252848
\][/tex]
7. Round the result: Round the result to three decimal places.
[tex]\[
0.0252848 \approx 0.025
\][/tex]
Therefore, the concentration of the drug in the bloodstream after 5 minutes is approximately:
[tex]\[ \boxed{0.025} \][/tex]
So, the drug concentration [tex]\( = 0.025 \)[/tex]
