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Word Problems: Exponential Decay
In order to solve problems involving exponential decay, it is necessary to:
Exponential decay is generally applied to word problems that involve financial applications as well as those that deal with radioactive decay, medicine dosages, and population decline. To decay exponentially means that the topic being studied is decreasing in  proportion to the amount that was previously present. The following is an example of an exponential decay problem.
When doctors prescribe medicine, they must consider how much the drug’s effectiveness will decrease as time passes. If each hour a drug is only 95% as effective as the previous hour, at some point the patient will not be receiving enough medicine and must be given another dose. If the initial dose was 250 mg and the drug was administered 3 hours ago, how long will it take for the initial dose to reach a dangerously low level of 52 mg?
First, we will need to use the general exponential decay formula:
In the formula, represents the amount of medicine after time has passed. represents the initial amount of medicine. The constant a represents the rate of decay (and is always a number between 0 and 1), and t stands for time, which is in hours in this problem.
Now, we need to substitute known values for the variables in the formula. The problem asks how long it will take the initial dose to become dangerously low. Therefore, is 52 in this problem. is the initial dose which is 250 mg. The rate of decay is which will be converted to the decimal 0.95. Time t is what we are trying to find. So we have the following:

Finally we must solve the equation for time t. To do so, first divide both sides by 250 to simplify the equation.

Next we take the log of each side of the equation and bring down the exponent, t. For a reminder on taking the log of both sides as well as the properties of logs, please examine this companion lesson.

Now, to solve for time t, divide both sides by (log 0.95) to obtain the following:
Now we use a calculator to find the value for t
Checking our answer shows

A(t) = 250(0.95)30.61 = 52.00673226
In fact, t actually represents less than the number of hours required for the amount of drug left to go below 52 mg. This is because there is still slightly more than 52 mg left at time t = 30.61 hours, our rounded off answer. The amount of the drug left will go below 52 mg sometime AFTER 30.61 hours has passed.

Let's Practice

Question #1
AudioNuclear energy derived from radioactive isotopes can be used to supply power to space vehicles. Suppose that the output of the radioactive power supply for a certain satellite is given by the function: . In the function is measured in watts and t is time in days. After how many days will the output be reduced to 25 watts?

Question #2
Exponential decay is used in determining the age of artifacts. The process involves calculating the percentage of carbon-14 that remains in the artifact. Carbon-14 decays exponentially with a half-life [T½] of approximately 5715 years. Half-life is the time required for half of a sample to disintegrate. Therefore, after 5715 years, a given amount of carbon-14 will have decayed to half the original amount. This process is used on artifacts that are up to 80,000 years old. Artifacts older than that do not have enough carbon-14 to date age accurately.
In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found. Analysis showed that the scroll wrappings contained 76 % of their original carbon-14. Estimate the age of the Dead Sea Scrolls using the formula where the decay constant which equals -0.000121 for this particular half-life of 5715.

Try These
Question #1
AudioPre-historic cave paintings were discovered in a cave in France. The paint contained 12% of the original carbon-14. Estimate the age of the paintings given that the constant k = -0.000121.

Question #2
AudioA city finds its residents moving to the suburbs. Its population is declining according to this relationship: . Given that the original population for the city was 1,000,000, how long will it take for the population to decline to half its initial number?

As you can see, this type of problem requires that you write an exponential decay function based on given information. You must then correctly substitute given values for variables and solve the equation you obtain. In solving the equation you must convert the exponential equation to a log equation and correctly use various properties of logarithms. At the conclusion of the problem, you should always check for the reasonableness of your solution.

D Saye

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