The concentration [tex]C(t)[/tex] of a certain drug in the bloodstream after [tex]t[/tex] minutes is given by the formula [tex]C(t) = 0.04 \left(1 - e^{-0.2t}\right)[/tex].

What is the concentration after 5 minutes? Round to three decimal places.

Answer:
[tex]\text{drug concentration} = \boxed{}[/tex]



Answer :

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]