APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**DECAY WORD PROBLEMS:**

**Problem 2:**

**Answer:**

**Solution:**

What does it mean to say that a substance has a half-life of 300 years? It means that every 300 years half of the substances disappears. After the first 300 years, one-half remains, after the second 300 years only one of remains, after the third 300 years, only of remains and so forth.

Let's create some points that describe the experiment and plot these points.
Examine the graph to determine what type of function we are dealing with.
All the points will have the form ( number of years, percent of
substance remaining)

At time 0, 100% of the substance is present; the corresponding point is
After 300 years only 50% of the substance is present;
the corresponding point is
After 600 years only
of the substance is present; the corresponding
point is
After 900 years only
of the substance is present. After 1200 years only
of the substance is present; the corresponding point
is
.

When you plot the data, the curve looks exponential. Therefore, the
mathematical model is probably exponential The model looks something like

where
represents the percent of the initial substance
remaining after t years, *a* represents the initial percent of the substance
(either 100% or 1.00), *t* represents the numbers of years that have
passed, and *b* represents the decay constant based on a base of *e*.(Note
that you can use any base. The base e is chosen for standardization purposes
only).

We know that or 1.00. However, you can verify it in the equation
Let *t*=0 in the equation.

The equation is now modified:

We know that or 0.50 of the initial substance remains after 300
years. Another way of saying this is that
In the
above equation, replace
with 0.50 and replace *t*with 300.

Take the natural logarithm of both sides of the equation:

The model is or

How long will it take the sample to decay to 10% of its initial amount??
Just substitute 0.10 for
in the equation.

Take the natural logarithm of both sides of the equation.

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