27. Scientifically, the half-life of caffeine in a human body is about 6 hours. Sophie 2 cups of coffee 10 hours ago. How much is left in her system? Hint: The equation of half-life is \( y(t)=a e^{-k t} \). A. About 0.09 of the original amount About 0.19 of the original amount About 0.31 of the original amount About 0.38 of the original amount

To find out how much caffeine is left in Sophie's system after 10 hours, we can use the half-life equation \( y(t) = a e^{-kt} \), where: - \( y(t) \) is the amount of caffeine left in the system at time \( t \), - \( a \) is the initial amount of caffeine, - \( k \) is the decay constant (related to the half-life), - \( t \) is the time in hours. Given that the half-life of caffeine is 6 hours, we can find the decay constant \( k \) using the formula: \[ k = \frac{\ln(2)}{t_{\text{half-life}}} \] Substitute the given half-life of 6 hours into the formula to find \( k \): \[ k = \frac{\ln(2)}{6} \] Now, we can use the half-life equation to find the amount of caffeine left in Sophie's system after 10 hours. Substitute the values of \( a \) (initial amount of caffeine), \( k \) (decay constant), and \( t \) (time) into the equation: \[ y(10) = a e^{-k \cdot 10} \] Since Sophie had 2 cups of coffee 10 hours ago, the initial amount of caffeine \( a \) is 2 cups. Substitute this value into the equation: \[ y(10) = 2 e^{-\frac{\ln(2)}{6} \cdot 10} \] Now, we can calculate the amount of caffeine left in Sophie's system after 10 hours. Calculate the value by following steps: - step0: Calculate: \(2e^{-\frac{\ln{\left(2\right)}}{6}\times 10}\) - step1: Multiply the numbers: \(2e^{-\frac{5\ln{\left(2\right)}}{3}}\) - step2: Transform the expression: \(2\times 2^{-\frac{5}{3}}\) - step3: Calculate: \(2^{-\frac{2}{3}}\) - step4: Express with a positive exponent: \(\frac{1}{2^{\frac{2}{3}}}\) - step5: Simplify: \(\frac{1}{\sqrt[3]{2^{2}}}\) - step6: Rewrite the expression: \(\frac{1}{\sqrt[3]{4}}\) - step7: Multiply by the Conjugate: \(\frac{\sqrt[3]{4^{2}}}{\sqrt[3]{4}\times \sqrt[3]{4^{2}}}\) - step8: Simplify: \(\frac{2\sqrt[3]{2}}{\sqrt[3]{4}\times \sqrt[3]{4^{2}}}\) - step9: Multiply the numbers: \(\frac{2\sqrt[3]{2}}{2^{2}}\) - step10: Reduce the fraction: \(\frac{\sqrt[3]{2}}{2}\) The amount of caffeine left in Sophie's system after 10 hours is approximately 0.63 cups, which is about 0.31 of the original amount of 2 cups of coffee.