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## Log-exp problem sheet 4-15-13

The Caffeine Problem

**Introduction **
In this lesson, we explore the dynamics of caffeine in the body through the use of exponential functions.

Various foods and drinks popular around the world contain caffeine. Caffeine is an alkaloid compound

that comes from plants, including coffee, tea, kola nuts, mate, cacao and guarana. Many people drink

caffeine drinks because they like the taste of them, others for the physical effect of the caffeine. Most

people are aware of differences in the way they feel as a result of drinking caffeine, which stimulates the

central nervous system, the heart muscles, and the respiratory system. The way individuals interpret the

effects of caffeine as a stimulant varies widely. For many, the effect is pleasant and energizing, a "wake

up" or a "pick-me-up", and it can delay fatigue. For others, the effects are unpleasant. Laboratory tests

indicate that 1 to 3 cups of coffee can produce an increased capacity for sustained intellectual effort and

decrease reaction time, but may adversely affect tasks involving delicate muscular coordination and

accurate timing.

The effect of coffee is quite different from the effect of alcohol, for example, with regard to increase in

mental capacity; such an increase is not seen in persons intoxicated with alcohol. Caffeine is classified

as a stimulant, whereas alcohol is a depressant. These two general classes of drugs have very different

effects on the human body.

**Quantity of caffeine per drink **

The amount of caffeine in different drinks varies, and some also contain other alkaloids that act as

stimulants or relaxants. Thus, it is difficult to relate the amount of caffeine in a drink to the physical

effect it may have on your body. We list an average, approximate amount of caffeine in some drinks.

The caffeine levels in commercial sodas tend to be consistent. The caffeine levels in coffee and tea vary

widely according both to the plant and to processing, but these numbers give some idea of the caffeine

level.

In an 8oz cup of COFFEE:

[1]

**Goodman and Gilman's The Pharmacological Basis of Therapeutics**: A.G. Gilman, L. S. Goodman,

T.W. Rall, and F. Murad, Macmillan Publishing Co., NY, 1985.

**Physical responses to caffeine **

The effects of caffeine can only be felt when the caffeine is present in sufficient amounts. For most

people, from 32 to 200 mg of caffeine acts as a minor stimulant; these amounts have been shown to

speed up reactions in simple routine tasks in laboratory experiments. As noted above, the minor

stimulant effect is experienced by some people as pleasant and by others as unpleasant. Steadiness of the

hand has been shown to be worse after 200 mg of caffeine. More than 300 mg is enough to produce

temporary insomnia. 480 mg has been known to cause panic attacks in panic disorder patients. Amounts

of 5 to 10 g (5000-10,000 mg) of caffeine cause death.

**Elimination of caffeine from the body **
The two primary ways that chemicals are eliminated from the body are through filtration by the kidneys

and metabolism by enzymes from the liver. Our bodies eliminate caffeine primarily by the functioning

of the kidneys. The kidneys tend to filter out a constant proportion of a chemical that proportion

depending on the particular chemical and individual. In the “average person”, about 13% of the caffeine

in the body is eliminated each hour.

** **

1. A person starts the day by drinking 3 cups of coffee containing 130 mg of caffeine

**each**.

**A. **How much caffeine will there be in this person's body 1 hour later? 2 hours later? 3 hours later?

**B. **Find a formula for the amount of caffeine t hours later.

**C. **What is the long-term behavior (end behavior) of the function you have generated? Make a graph

**D. **How much caffeine will be in this person's body in 30 minutes?

**E. **How long will it take until the amount of caffeine in this person's body is cut in half?

**F**. Rewrite the function from part

**B** incorporating the “half-life” you computed in part

**E**.

**2.** Consider the following typical caffeine day for Mr. Jones:

**Caffeine in drink **
In the following, assume there was no caffeine in Mr. Jones' system prior to 7:00 AM.

**A**. Create a function with the following structure:

Let’s assume the domain of the function is [0, 24] with the understanding that Jones does NOT drink

any more coffee after 5:00 PM.

**B**. What is the maximum caffeine level in Mr. Jones' system? Justify your answer!

**C**. Assuming Mr. Jones does NOT drink any more coffee the next day, when will his caffeine level first

drop below 50 mg?

**D**. If Mr. Jones has the same caffeine intake on the following day, how would your function have to be

adjusted (so that the domain is now [0, 48])?

**E**. If Mr. Jones has the same caffeine intake every day from this point on, what would his caffeine level

be after 7 full 24 hour cycles (just before his before his final 7:00 AM coffee?) After

*n* full cycles? You

are invited to use some estimation here.

**3. ** Mr. Jones' car is worth $10,000 at the beginning of this year. Its value is depreciating at a rate of

10% per year. However, at the beginning of next year, he intends to replace the engine which will

increase the value by $2000; and the year after that, he intends to replace the transmission which will

increase the value by $1500.

**A**. Create a piece-wise defined function in which the input (independent variable) is time (in years after

2006) and the output is the value of Jones' car at that time.

**B**. Make a careful sketch of the function from part (A).

**C**. Find the first and the second time that Jones' car's value will be $9500. Express your answer in years

(rounded to two places after the decimal place).

**4. **A mutant strain of bacteria is on the loose. Its population doubles every 2.5 hours. At 12:00 noon

today, the population was 5000.

**A.** Build a function that measures the population of the bacteria as a function of time (in hours after

12:00 noon today).

**B. **To the nearest minute, find the clock time at which the population reaches 100,000.

**C**. In one hour, by what percent

*(rounded to the nearest hundredth of a percent)* does the population

grow?

**D. **If your formula from part (A) is converted to

form, then what is the value of

*k*?

*(Here, I want an exact answer)* * *

**5. **Mr. Jones is slowing down. He has determined that his vertical leap decreases by 5 percent each year.

Currently, he has a standing vertical leap of 20 inches.

**A. **Build a function that measures Mr. Jones' vertical leap (in inches) as a function of time (in years since

2006).

**B.** The Sunday NY Times is usually about 8 inches high. Assuming the function from part (A) holds for

the future, when will Mr. Jones no longer be able to jump higher than a Sunday Times?

*(Round to the *

nearest hundredth of a year

**C. **After practicing his vertical leap all day, Mr. Jones is sore. Luckily he has his special

microwaveable heating pad to apply to his aching back. When he takes the pad out of the microwave,

its temperature is

. Three minutes later the temperature has dropped to
apartment) while its temperature is between
long will he be able to use it before having to put it back into the microwave?

*[Round your final answer to the nearest hundredth of a minute]*
**Newton’s Law of Cooling** is based on the fact that the rate of change in the temperature of an object is

proportional to the difference between the temperature of the object and the external ambient

temperature.

The formula we just developed in class is:

*F* = the temperature of the object (this is a variable)
A = the ambient (surrounding) temperature (this is a constant)
= the initial temperature of the object (this is a constant)

*k* = a constant particular to the cooling properties of the object in question

**2. ** On a Wednesday, the body of Mr. Jones is discovered by in his office by the Ultimate Team at

3:30 PM (they came looking fro him when he didn’t show up for practice).

At the time of discovery, the temperature of Jones’ body is

15 minutes later, the temperature of Jones’ body has dropped to
It is common knowledge Mr. Jones keeps his office at a constant temperature of
As the ultimate team wants to avenge Mr. Jones’s death, they bust out their TI-84’s, consult the Math Dept Master schedule (see below) and determine that ______________ is the murderer.

Source: http://courses.horacemann.org/pluginfile.php/77236/mod_page/content/1/Caffeine_problems_4-15-13.pdf

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