Question

The half-life of caffeine is 5 hours. If you ingested a 30 oz Big Gulp, after how much time will have to pass before you have under 1 oz of caffeine remaining?

Answer 1

Answer:

The time that will have to pass before one has under 1 oz of caffeine remaining is 24.53 hours

Explanation:

Here, we have the formula for half life as follows;

$N(t) = N_0(\frac{1}{2})^{\frac{t}{t_{1/2}}$

Where:

N(t) = Remaining quantity of the substance = 1 oz

N₀ = Initial quantity of the substance = 30 oz

t = Time duration

$t_{1/2}$ = Half life of the substance = 5 hours

Therefore, plugging in the values, we have

$1= 30(\frac{1}{2})^{\frac{t}{5}}$

$\frac{1}{30} = (\frac{1}{2})^{\frac{t}{5}}\\ln(\frac{1}{30}) =\frac{t}{5} ln(\frac{1}{2})\\\frac{t}{5} = \frac{ln(\frac{1}{30}) }{ ln(\frac{1}{2})} = 4.91\\ t = 4.91 \times 5 = 24.53 \ hours$

The time that will have to pass before one has under 1 oz of caffeine remaining = 24.53 hours.