APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

DECAY WORD PROBLEMS:

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.

Example 3: The radioactive isotope sodium-24 is used as a tracer to measure the rate of flow in an artery or vein. The half-life of sodium-24 is 14.9 hours. Suppose that a hospital buys a 40-g sample of sodium-24 and will reorder when the sample is reduced to 3 g.

• a: How much of the sample will remain after 50 hours?.

• b: How often before the hospital has to reorder sodium-24?

• c: How much of the sample will remain after 1 year?

Solution and Explanation:

First, what does it mean to say that the half-life of sodium-24 is 14.9 hours? It means that after 14.9 hours only half of the original amount remains. After another 14.9 hours one-half of that one-half amount remains. Another way of saying that is that after 29.8 hours only of or of the original amount remains. Make a table showing the relationship between the number of hours that have passes and the amount of sodium-24 remaining.

Let's us form the equation with base e:

At time 0, the hospital had 40 g. We can say the same thing with the equation

The equation is now

After 14.9 hours,there is only of 40=20 g left. Another way of say this is

Take the natural log of both sides of the equation.

The equation can be written as

The decay constant is

How much of the sodium-24 will remain after 50 hours? Just replace t in the formula with 50.

How often before the hospital has to reorder sodium-24? or How long will it before the sample is reduced to 3 g? Replace with 3 g and solve for t.

Take the natural logarithm of both sides of the equation.

How much of the sample will remain after 1 year?

First convert 1 year to hours.

Substitute 8,760 for t in the equation .

This is an extremely small number.

If you would like to work another example, click on example

If you would like to test your knowledge by working some problems, click on problem.

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