Question

the concentration ( c(t) ) of a certain drug in the bloodstream after ( t ) minutes is given by the formula ( c(t) = 0.02(1 - e^{-0.2t}) ). what is the concentration after 5 minutes? round to three decimal places.

Explanation:

Step1: Substitute t = 5 into the formula

We have the formula \( C(t) = 0.02(1 - e^{-0.2t}) \). Substitute \( t = 5 \) into it, so we get \( C(5)=0.02(1 - e^{-0.2\times5}) \).

Step2: Calculate the exponent

First, calculate the exponent part: \( -0.2\times5=-1 \), so the formula becomes \( C(5)=0.02(1 - e^{-1}) \).

Step3: Calculate \( e^{-1} \)

We know that \( e^{-1}=\frac{1}{e}\approx\frac{1}{2.71828}\approx0.36788 \).

Step4: Calculate the value inside the parentheses

Then, calculate \( 1 - e^{-1}\approx1 - 0.36788 = 0.63212 \).

Step5: Calculate the final concentration

Multiply this result by 0.02: \( C(5)=0.02\times0.63212 = 0.0126424 \).

Step6: Round to three decimal places

Rounding 0.0126424 to three decimal places, we look at the fourth decimal place which is 6, so we round up the third decimal place. Thus, \( 0.0126424\approx0.013 \).

Answer:

\( 0.013 \)