Precalculus Exponential and Logarithmic Functions Applications
Growth and Decay Concepts 1

|  |  | Definition 1 |  |
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Exponential Growth or Decay Any quantity that always changes by a fixed multiple over the same increment of time satisfies the formula A(t) = A 0 · e k·t , where A(t) denotes the amount of the quantity at time t, A 0 is the amount at time 0, and k is a constant. Such quantities are said to grow (k > 0) or decay (k < 0) exponentially.

|  |  | Definition 2 |  |
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The half-life of a radioactive substance is the time it takes for exactly half of the initial quantity to decay.

Example 1. When a bactericide is introduced into a culture of bacteria, the number of bacteria , A(t), is approximated by A(t) = 250,000·e-0.4·t ,

where t is time measured in hours. Find the time it will take until only 25,000 bacteria are present. Solution We must solve the equation
25,000 = 250,000·e-0.4·t
for t.
0.1 = e-0.4·t
-0.4·t = ln0.1
Use a calculator.
t = 5.76

Answer: t = 5.76 hours.

Example 2. If 100 grams of radium present initially and the amount of radium remaining after 25 years is 98.9 grams, determine the half-life of radium.

Solution
First, we use the given information to evaluate the constant k. A(25) = 98.9, A0 = 100.
98.9 = 100·ek·25
0.989 = e25·k
25·k = ln0.989
k = 0.04·ln0.989
k = 0.04·(-0.0110609)
k = -0.00044244
We must find the time when A(t) = 0.5·A0, that is, we must solve the equation
0.5·A0 = A0·e-0.00044244·t
for t.
0.5= e-0.00044244·t
-0.00044244·t = ln0.5
Use a calculator.
t = 1,567

Answer: t = 1,567 years.

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