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 amount of cells 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 are 100,000 cells, and the corresponding point is (0, 100,000). At t = 1, there are 50,000 cells, and the corresponding point is (1, 50,000). At t = 2, there are 25,000 cells, and the corresponding point is (2, 25,000). At t = 3, there are 12,500 cells, and the corresponding point is (3, 12,500).
You could also say that after 1 minute the population was
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You could say that after 2 minutes, the population was
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After 3 minutes the population was
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The population formula is therefore
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Example 8: Determine the number of cells after 10 minutes:
Solution and Explanation:
Substitute 10 for t in the equation tex2html_wrap_inline163 :
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Since you cannot have part of a cell, it appears that the population has declined to 98 cells after ten minutes.
Example 9: Determine how long it would take the population (number of cells) to reach 10 cells.
Solution and explanation:
Here you know the number of cells at the beginning of the study (100,000) and you know the number of cells at the end of the study (1), but you do not know the time. Substitute 10 for f(x) in the equation tex2html_wrap_inline163 :
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Divide both sides by 100,000:
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Take the natural logarithm of both sides:
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Simplify the right side of the equation using the third rule of logarithms:
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Divide both sides by tex2html_wrap_inline177 and simplify:
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It would take a little more than 13 minutes for the population of cells to reach 10.
Example 10: Write an equation with base e that is equivalent to the equation
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Solution and Explanation:
Let's start with a generic exponential equation with base e:
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The f(t) represents the size of the population (number of cells) at time t, the t represents the time, and the a and b represent adjusters when we change the base. The value of a is also the size of the population at the start of the study (t = 0), and b is the relative growth rate with respect to the base e. Before we even calculate their values, let's see if we can determine a few things about a and b.
The starting population is 100,000, so the value of a should be 100,000. The value of b should be negative, because b is the growth rate and the population is declining. Let's see if we are right.
We know that the population is 100,000 at time 0, so insert these numbers in the equation
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We have
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We now know that the value of a in the adjusted equation is 100,000.
Rewrite the equation with a = 100,000:
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We know that the population after 1 minute is 50,000 cells, so insert these numbers in the equation tex2html_wrap_inline221 :
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Divide both sides of the equation by 100,000:
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Solve for b by taking the natural logarithm of both sides of the equation tex2html_wrap_inline229 .
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Simplify the right side of the equation using the third rule of logarithms:
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Simplify the left side of the equation:
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Insert this value of b in the equation
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and the equation is simplified to
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We know that the population is 25,000 after 2 minutes, so use these values to check the validity of the above equation. Substitute 2 for t in the right side of the above equation. If the answer is 25,000, then the model is correct.
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The model (equation) is correct.
We know from the original equation that after 3 seconds, the population is 12,500. Let's do a second check.
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The exponential equation
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with base e is equivalent to the exponential equation
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with base tex2html_wrap_inline153 . Note that the natural log of the base tex2html_wrap_inline153 is also the relative growth rate when the base is e.
By now you may have concluded that in the equation
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the 100,000 represents the value at the start of the study (t = 0) and the -0.693147 represents the relative rate of growth or decline with respect to the base e.
