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

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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.
About 0.31 of the original amount is left in Sophie's system.

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