Math  /  Algebra

QuestionThe half-life of caffeine in the human body is about 6.4 hours. A cup of coffee has about 100 mg of caffeine. a. Write an equation for the amount of caffeine in a person's body after drinking a cup of coffee? Let CC be the milligrams of caffeine in the body after tt hours. \square b. How much caffeine will remain after 10 hours? \square mg c. How long until there are only 20 mg remaining? \square hours

Studdy Solution
To find the time when only 20 mg remains, set C(t)=20 C(t) = 20 and solve for t t :
20=100×(12)t6.4 20 = 100 \times \left(\frac{1}{2}\right)^{\frac{t}{6.4}}
Divide both sides by 100:
0.2=(12)t6.4 0.2 = \left(\frac{1}{2}\right)^{\frac{t}{6.4}}
Take the logarithm of both sides:
log(0.2)=t6.4log(12) \log(0.2) = \frac{t}{6.4} \log\left(\frac{1}{2}\right)
Solve for t t :
t=6.4×log(0.2)log(0.5) t = 6.4 \times \frac{\log(0.2)}{\log(0.5)}
Calculate the value:
t6.4×2.3219 t \approx 6.4 \times 2.3219
t14.86 hours t \approx 14.86 \text{ hours}
The solutions are: a. C(t)=100×(12)t6.4 C(t) = 100 \times \left(\frac{1}{2}\right)^{\frac{t}{6.4}} b. 34.1 34.1 mg c. 14.86 14.86 hours

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