APPLICATIONS OF EXPONENTIAL | |

AND | |

LOGARITHMIC FUNCTIONS |

**DECAY WORD PROBLEMS:**

**Problem 3:**

**Answer:**

**Solution:**

Since the population is declining, you know that the growth factor is
negative and is called the decay factor or the decay constant. Since we are
assuming an exponential model, it means that an exponential equation will
describe the size of the population at any time between 1945 and 1996. The
generic population model is

We need to solve for the value of a and the value of b.

We started examining the population size in 1945, so let *t*=0 in 1945. This
means that when *t*=0, the population size was 120,000. A mathematical way
of saying this is ,
Substitute these values in
the above equation.

The equation can now be rewritten as

We know that in 1996 (
*t*=1996-1945=51), the population was 65,000. A
mathematical way of saying this is

Let's solve for b.

Take the natural logarithm of both sides of the equation.

The equation describing the population size at any years between 1945 and
1996 is

In 1960,
*t*=1960-1945=15. To find the population in 1960, simply substitute
15 for t in the above equation.

The population in 1960 was about 100,200.

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