Caffeine occurs naturally in a variety of food products such...

Caffeine occurs naturally in a variety of food products such as coffee, tea, and chocolate. The kidneys filter the blood and remove caffeine and other drugs through urine. The biological half-life of caffeine is approximately $6 \mathrm{hr}$. If one cup of coffee has $80 \mathrm{mg}$ of caffeine, then the amount of caffeine $C$ (in $\mathrm{mg}$ ) remaining after $t$ hours is given by $C=80(2)^{-t / 6}$. a. How long will it take for the amount of caffeine to drop below $60 \mathrm{mg}$ ? Round to 1 decimal place. b. Laura has trouble sleeping if she has more than $30 \mathrm{mg}$ of caffeine in her bloodstream. How many hours before going to bed should she stop drinking coffee? Round to 1 decimal place.

Caffeine occurs naturally in a variety of food products such as coffee, tea, and chocolate. The kidneys filter the blood and remove caffeine and other drugs through urine. The biological half-life of caffeine is approximately $6 \mathrm{hr}$. If one cup of coffee has $80 \mathrm{mg}$ of caffeine, then the amount of caffeine $C$ (in $\mathrm{mg}$ ) remaining after $t$ hours is given by $C=80(2)^{-t / 6}$.
a. How long will it take for the amount of caffeine to drop below $60 \mathrm{mg}$ ? Round to 1 decimal place.
b. Laura has trouble sleeping if she has more than $30 \mathrm{mg}$ of caffeine in her bloodstream. How many hours before going to bed should she stop drinking coffee? Round to 1 decimal place.

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Key Concepts

Solving Exponential Inequalities

Solving exponential inequalities involves determining the time at which an exponentially decaying function becomes less than a certain threshold. This process typically combines understanding of exponential functions with logarithmic manipulation to find when the concentration drops below a specified limit.

Logarithms in Exponential Equations

Logarithms are used to solve equations where the variable is an exponent, which is common in models of exponential decay. They enable the isolation of the time variable in the decay formula, allowing for the calculation of how long it takes for a substance's amount to fall below a certain level.

Biological Half-Life

The biological half-life is the time required for a substance within the body to reduce to half its initial concentration due to physiological processes like metabolism and excretion. This concept is crucial for understanding the rate at which caffeine is eliminated from the bloodstream.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In this context, it explains how the caffeine concentration in the blood decreases over time based on a constant half-life.

Transcript

00:01 So in this expression, what i have is after you drink a cup of coffee, you have 80 milligrams of caffeine, and then after tea hours is the amount of caffeine in your system.

00:10 So first, how long will it take for the amount of caffeine to drop below 60 milligrams? okay, so i want this to be less than 60.

00:17 So i need my caffeine to be 60 equals 80 -t negative zero.

00:24 T over six.

00:25 What do i do first? i want to get my t -term by itself, so i'm going to divide by 80 to both sides.

00:30 So 60 divided by 80, that's approximately 0 .75 or 3 fourths equals 2, negative t over 6.

00:38 How do i get rid of that 2? well, how do i get rid its exponent? what i could do is just take the natural log of both sides.

00:45 I could also take the log base 2, or i could take the natural log.

00:49 This will be ln of 0 .75.

00:52 Rules of logarithms bring down the negative t over 6 times ln of 2.

00:58 And then what i could do is multiply by the, i'm trying to get t by itself, so the reciprocal of this fraction 6 over negative l and of 2 to both sides, times 6 over negative l and of 2.

01:15 This cancels with the negative especially.

01:18 So t, if i were to plug this in my calculator times l &o 0 .75 divided by negative l and of 2.

01:26 How much time would it take? 2 .49 hour...

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