APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**DECAY WORD PROBLEMS:**

**Problem 1:** If you start a biology experiment with 5,000,000 cells
and half the cells are dying every 10 minutes, how long will it take to
have less than 1,000 cells?

**Answer:** 123 minutes.

**Solution:**

At time 0, there are 5,000,000 cells. At time 10 minutes, there are
0.50 x 5,000,000 = 2,500,000 cells remaining. At time 20
minutes, there are 0.50 x 2,500,000 = 1,250,000 cells remaining. At time 30 minutes, there are
0.50 x 1,250,000 = 625,000 cells remaining. When you plot the data, the
curve looks exponential. Therefore, the mathematical model is probably
exponential The model looks something like

where
represents the number of cells remaining after t
minutes of observation, *a* represents the number of cells at the start of
the experiment (5,000,000) , *t* represents the numbers of
minutes since the experiment began, and *b* represents the decay constant
based on a base of *e*.

We know that
*a*=5,000,000 because we started with five million cells.
However, you can verify it in the equation
Let *t*=0 in the equation.

The equation is now modified:

We know that there are 2,500,000 cells after 10 minutes. Another way of
saying this is that
In the above equation,
replace
with 2,500,000 and replace *t* with 10.

Take the natural logarithm of both sides of the equation:

The equation describing the number of cells remaining after a certain
number of minutes is

Let's check it out by seeing if this model will give us 1,250,000 cells
after twenty minutes.

The model is

How long will it take the sample to decay to below 1,000 cells? Just
substitute 1,000 for
in the equation.

Take the natural logarithm of both sides of the equation.

It will take about 123 minutes for the cell population to drop below a 1,000
count.

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