(Population Word Problems)
To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.
In this section, we will review population problems. We will also discuss why the base of e is used so often with population problems.
Example 1: Suppose that you are observing the behavior of cell duplication in a lab. In one experiment, you started with one cell and the cells doubled every minute. Write an equation with base 2 to determine the number (population) of cells after one hour.
Solution and Explanations:
First record your observations by making a table with two columns: one column for the time and one column for the number of cells. The number of cells (size of population) depends on the time. If you were to graph your findings, the points would be formed by (specific time, number of cells at the specific time). For example, at t = 0, there is 1 cell, and the corresponding point is (0, 1). At t = 1, there are 2 cells, and the corresponding point is (1, 2). At t = 2, there are 4 cells, and the corresponding point is (2, 4). At t = 3, there are 8 cells, and the corresponding point is (3, 8).
It appears that the relationship between the two parts of the point is exponential. At time 0, the number of cells is 1 or 20 = 1. After 1 minute, when t = 1, there are two cells or 21 = 2. After 2 minutes, when t = 2, there are 4 cells or 22 = 4.
Therefore, one formula to estimate the number of cells (size of population) after t minutes is the equation (model)
Determine the number of cells after one hour:
f (60) = 260 = 1.15×1018
Example 2: Determine how long it would take the population (number of cells) to reach 100,000 cells.
Solution and explanation:
It would take 16.6 minutes, rounded, for the population (number of cells) to reach 100,000.
Example 3: Write an equation with base 5 that is equivalent to the equation
Solution and Explanation:
rounded to 0.4307.
If you would like to work another example, click on Example
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.
Author: Nancy Marcus