Question

The half life of caffeine in a healthy adult is 5.7 hours. Jeremiah drinks 16 ounces of caffeinated coffee in the morning. How long will it take for only 40% of the caffeine to remain in his body?

Answer 1

It will take 7.5 hours for only 40% of the caffeine to remain in his body.

Step-by-step explanation:

Half-life (symbol t1⁄2) is the time required for a quantity to reduce to half of its initial value.

The half-life of caffeine is 5.7 hours.

It means that if we have 10 ounces of caffeine. After 5.7 hours, the remaining caffeine will be equal to 5 ounces and so on.

And the decaying speed depends on the initial amount of the substance.

In the given question.

t1⁄2 = 5.7 hours

Initial amount = N(i) = 16 ounces

Remaining amount after time t = N(t) = 40% of 16 = 6.4 ounces

time t = ?

Using the following formula for remaining amount of substance after time t:

N(t) = N(i)*(0.5)^(t/t1⁄2)

we can find the time t

putting the values in the formula given above, we get:

$6.4 = 16(\frac{1}{2} )^\frac{t}{5.7}\\ \frac{6.4}{16} = (0.5)^\frac{t}{5.7}\\ 0.4=(0.5)^\frac{t}{5.7}$

Taking natural log on both sides:

$ln(0.4) = ln(0.5)^\frac{t}{5.7}\\ln(0.4) = \frac{t}{5.7}(ln(0.5))\\t = \frac{ln(0.4)}{ln(0.5)}5.7\\t= 1.32*5.7\\t=7.5 hours$

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