Aimee is studying bacterial growth in the laboratory. She starts her experiment with 1 bacterium and then counts the bacteria at regular intervals and records them in the table below. If the pattern continues, how long will it take to have over 1000 bacteria?
[tex]\center\begin{array}{|c|c|c|c|c|}\cline{1-5}\sf Time \ (hours) & \sf 0 & \sf 3 & \sf 6 & \sf 9\\\cline{1-5}\sf Number \ of \ Cells & \sf 1 & \sf 2 & \sf 4 & \sf 8\\\cline{1-5}\end{array}[/tex]
[tex]\center\begin{array}{|c|c|c|c|c|}\cline{1-5}\sf Time \ (hours) & \sf 0 & \sf 3 & \sf 6 & \sf 9\\\cline{1-5}\sf Number \ of \ Cells & \sf 1 & \sf 2 & \sf 4 & \sf 8\\\cline{1-5}\end{array}[/tex]
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Given: Recordings of observations of bacteria.
To find: How long it would take to have 1000 bacteria.
Answer:
Let's consider the recordings of the number of cells.
1, 2, 4, 8, ... .
This can be written as 2⁰, 2¹, 2², 2³, ... .
Let's assume the powers to be 'n'.
first crosses 1000 at 1024, where 'n' is 10.
Let's now observe the recording of the hours.
0, 3, 6, 9, ... .
Each time, the hour is increased by 3.
Through this, we understand that
At 0 hours, we have 1 (2⁰) bacteria.
At 3 hours, we have 2 (2¹) bacteria.
At 6 hours, we have 4 (2²) bacteria.
At 9 hours, we have 8 (2³) bacteria.
At 12 hours, we have 16 (2⁴) bacteria.
At 15 hours, we have 32 (2⁵) bacteria.
At 18 hours, we have 64 (2⁶) bacteria.
At 21 hours, we have 128 (2⁷) bacteria.
At 24 hours,we have 256 (2⁸) bacteria.
At 27 hours, we have 512 (2⁹) bacteria.
As mentioned earlier,
first crosses 1000 at 1024 where 'n' is 10.
Therefore, after 30 hours, we'll have over 1000 (1024 ⇒ 2¹⁰) bacteria.