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 2:** Hospitals utilize the radioactive substance
iodine-131 in the diagnosis of conditions of the thyroid gland. The
half-life of iodine-131 is eight days.

- a: Determine the decay constant b.

- b: If a hospital acquires 2 g of iodine-131, how much of this
sample will remain after 20 days ?

- c: How long will it be until only 0.01 g remains?

**Solution and Explanation:**

First, what does it mean to say that the half-life of iodine-131 is eight
days ?

It means that after eight days only half of the original amount remains.
After another eight days one-half of that amount remains. Another way of
saying this is that after 16 days only
of
or
of the original amount remains. Make a table showing the
relationship between the number of days that have passes and the remaining
amount of the iodine-131.

Let's us form the equation with base *e*:

At time 0, the hospital had 2 g. We can say the same thing with the equation

The equation is now

After 8 days, there is only 1 g left. Another way to say this is

Take the natural logarithm of both sides of the equation what you started The model now can be written as

The decay constant is

How much of the iodine-131 will remain after 20 days? Just replace *t* in
the equation with 20.

After 20 days, there will be 0.353556 g left.

How long will it be until only 0.01 g remains? Replace
with 0.01 g and solve for *t*.

Take the natural logarithm of both sides of the equation.

It would take a little over 61 days for the sample to be reduced to 0.01
g. To be more specific, it would take

It would take 61 days, 3 hours, and about 37 minutes for the 2 g sample to
be reduced to 0.01 g.

Let's us check this answer

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