6. Soon after taking an aspirin tablet a patient has absorbed 300 mg of the drug. If half of the aspirin in the bloodstream is removed every two hours, how much aspirin will be in the bloodstream after 5 hours? When will the amount of aspirin in the bloodstream be 100 mg ?

Let the amount of aspirin in the bloodstream after \( t \) hours be given by \[ A(t) = 300 \left(\frac{1}{2}\right)^{t/2}. \] ### Part 1: Amount after 5 Hours Substitute \( t = 5 \) into the formula: \[ A(5) = 300 \left(\frac{1}{2}\right)^{5/2}. \] This can be left in exponential form or approximated numerically. **Exact form:** \[ A(5) = \frac{300}{2^{5/2}}. \] **Numerical approximation:** Since \[ 2^{5/2} = 2^{2.5} = 4\sqrt{2} \quad \text{and} \quad \sqrt{2} \approx 1.414, \] we have \[ 2^{5/2} \approx 4 \times 1.414 \approx 5.656. \] Thus, \[ A(5) \approx \frac{300}{5.656} \approx 53\,\text{mg}. \] ### Part 2: Time when Aspirin Amount Is 100 mg We need to solve for \( t \) when \[ 300 \left(\frac{1}{2}\right)^{t/2} = 100. \] Divide both sides by 300: \[ \left(\frac{1}{2}\right)^{t/2} = \frac{100}{300} = \frac{1}{3}. \] Take the natural logarithm of both sides: \[ \frac{t}{2} \ln\left(\frac{1}{2}\right) = \ln\left(\frac{1}{3}\right). \] Solve for \( t \): \[ t = \frac{2 \ln\left(\frac{1}{3}\right)}{\ln\left(\frac{1}{2}\right)}. \] Recognize that \[ \ln\left(\frac{1}{3}\right) = -\ln 3 \quad \text{and} \quad \ln\left(\frac{1}{2}\right) = -\ln 2, \] thus \[ t = \frac{2(-\ln 3)}{-\ln 2} = \frac{2 \ln 3}{\ln 2}. \] **Numerical approximation:** Using \( \ln 3 \approx 1.0986 \) and \( \ln 2 \approx 0.6931 \), \[ t \approx \frac{2 \times 1.0986}{0.6931} \approx \frac{2.1972}{0.6931} \approx 3.17\,\text{hours}. \] ### Final Answers - The amount of aspirin in the bloodstream after 5 hours is \[ A(5) = 300 \left(\frac{1}{2}\right)^{5/2} \approx 53\,\text{mg}. \] - The aspirin amount will be 100 mg when \[ t = \frac{2\ln 3}{\ln 2} \approx 3.17\,\text{hours}. \]