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.

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