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
displaymath155
You could say that after 2 minutes, the population was
displaymath157
After 3 minutes the population was
displaymath159
The population formula is therefore
displaymath161
Example 8: Determine the number of cells after 10 minutes:
Solution and Explanation:
Substitute 10 for t in the equation tex2html_wrap_inline163 : displaymath165
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 : displaymath169
Divide both sides by 100,000: displaymath171
Take the natural logarithm of both sides: displaymath173
Simplify the right side of the equation using the third rule of logarithms: displaymath175
Divide both sides by tex2html_wrap_inline177 and simplify: displaymath179
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
displaymath183
Solution and Explanation:
Let's start with a generic exponential equation with base e: displaymath187
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 displaymath213
We have
displaymath215
We now know that the value of a in the adjusted equation is 100,000.
Rewrite the equation with a = 100,000: displaymath219
We know that the population after 1 minute is 50,000 cells, so insert these numbers in the equation tex2html_wrap_inline221 : displaymath223
Divide both sides of the equation by 100,000: displaymath225
Solve for b by taking the natural logarithm of both sides of the equation tex2html_wrap_inline229 . displaymath231
Simplify the right side of the equation using the third rule of logarithms: displaymath233
Simplify the left side of the equation: displaymath235
Insert this value of b in the equation displaymath239
and the equation is simplified to
displaymath241
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. displaymath245
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. displaymath247
The exponential equation
displaymath241
with base e is equivalent to the exponential equation
displaymath183
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
displaymath261
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.
32 Marcus has created a budget for his upcoming trip to the theme park. Admission is 40% of the budget. He plans to spend 32% of his money on food, 23% on
