Five bacteria are placed in a dish and then counted each hour. The first three bacteria counts were 12, 26, 54. If the pattern continues, how many hours after the bacteria were placed in the dish will there be 222 bacteria?
looks like
Tn+1 = 2Tn + 2
so, write down the next few terms
To determine the pattern, let's calculate the growth rate for each hour by finding the difference between consecutive counts:
The growth rate from the first to the second hour is: 26 - 12 = 14.
The growth rate from the second to the third hour is: 54 - 26 = 28.
From these calculations, we can see that the growth rate is increasing by 14 each hour.
Now, we need to determine how many times the growth rate of 14 needs to be added to the last count of 54 to reach 222.
Let's set up an equation to solve for the number of hours:
54 + 14x = 222
Subtract 54 from both sides of the equation:
14x = 222 - 54
14x = 168
Divide both sides of the equation by 14:
x = 168 / 14
x = 12
Therefore, it will take 12 additional hours to reach a count of 222 bacteria.
We can see that the growth rate of the bacteria is increasing over time. To find the pattern, let's calculate the differences between consecutive counts:
1st difference: 26 - 12 = 14
2nd difference: 54 - 26 = 28
We can observe that the 2nd difference is double the size of the 1st difference. This suggests that the bacteria count is growing exponentially.
To find a general formula for the bacteria count at any given hour, we will consider the growth rate. The pattern indicates that the growth rate is doubling each hour. So, for each hour, the bacteria count is multiplied by 2.
Now, let's write out the pattern after the 3rd hour:
1st hour: 12 bacteria
2nd hour: 12 * 2 = 24 bacteria
3rd hour: 24 * 2 = 48 bacteria
4th hour: 48 * 2 = 96 bacteria
5th hour: 96 * 2 = 192 bacteria
6th hour: 192 * 2 = 384 bacteria
We can see that after 6 hours, the bacteria count exceeds 222 bacteria. Therefore, the answer is 6 hours.