Bacteria are the most common example of exponential growth. Select a number
between 2 and 10 to represent the hourly growth rate of a certain bacteria. For
example, selecting the number 8 would mean that the amount of bacteria will be 8
times greater after every hour.
a. Suppose you start with one single bacterium. Make a table of values showing
the number of bacteria that will be present after each hour for the first six
hours using the hourly growth rate that you selected. Then determine how
many bacteria will be present once 24 hours have passed.
b. Explain why this table represents exponential growth.
c. Using this example, explain why any nonzero number raised to a power of
zero is equal to one.
d. Write a rule for this table.
e. Suppose you started with 100 bacteria, but they still grew by the same
growth factor. How would your rule change? Explain your answer.
a) Let's say we select the number 3 as the hourly growth rate. The table of values showing the number of bacteria present after each hour for the first six hours is as follows:
Hour | Number of Bacteria
------------------------
1 | 1
2 | 3
3 | 9
4 | 27
5 | 81
6 | 243
To determine the number of bacteria present after 24 hours, we can continue this pattern of exponential growth:
Hour | Number of Bacteria
------------------------
24 | 1 * (3^24) = 28,998,881
So, after 24 hours, there will be approximately 28,998,881 bacteria present.
b) This table represents exponential growth because the number of bacteria is increasing rapidly over time, with each hour's growth being multiplied by the growth rate. The relationship between the number of hours and the number of bacteria follows a pattern where each increment in the hour results in a greater increment in the number of bacteria.
c) In this example, any nonzero number raised to a power of zero is equal to one because according to the rule of exponential growth, when the exponent is zero, the result is always one. This can be understood by observing the pattern in the table: for the first hour (exponent 1), we have 1 * (3^1) = 3; for the second hour (exponent 2), we have 1 * (3^2) = 9; and so on. As we can see, when the exponent is equal to zero, we have 1 * (3^0) = 1, which implies that any nonzero number raised to the power of zero is equal to one.
d) The rule for this table can be written as: Number of Bacteria = 1 * (3^Hour), where "Hour" represents the number of hours passed.
e) If we start with 100 bacteria and they still grow by a factor of 3, the rule would change to: Number of Bacteria = 100 * (3^Hour). This is because the initial amount of bacteria is 100, and each hour's growth rate needs to be multiplied by this initial amount. So, the growth becomes proportional to the initial amount of bacteria, resulting in a higher number of bacteria after each hour.