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 selected the growth rate of 5.
Time (in hours) | Number of Bacteria
----------------------------------
1 | 1
2 | 5 (1 x 5)
3 | 25 (5 x 5)
4 | 125 (25 x 5)
5 | 625 (125 x 5)
6 | 3125 (625 x 5)
After 24 hours, the number of bacteria would be: 1 x 5^24 = 1 x 7,625,597,484,987,000
b. This table represents exponential growth because the number of bacteria is multiplying by the same growth factor (in this case, 5) at each time interval. The growth is not linear, but rather gets faster and faster as time goes on.
c. In this example, after 0 hours, there is still one bacterium present. This means that 5^0 should be equal to 1. When any nonzero number is raised to the power of zero, it is defined to be equal to one.
d. The rule for this table is: Number of bacteria = (Initial number of bacteria) x (Growth rate)^Time
e. If we started with 100 bacteria, the rule would change to: Number of bacteria = (Initial number of bacteria) x (Growth rate)^Time. The only difference is the initial number of bacteria, which is now 100 instead of 1.