Question
A biologist is researching a newly discovered bacteria that doubles in number every hour. You can use a function to describe the number of bacteria after x hours in a sample that originally contained 100 bacteria. Is the function linear or exponential? Which equation represents the function?
Answers
GPT-4o mini
The growth of the bacteria described in the scenario is exponential, not linear. This is because the population doubles at regular intervals (every hour), which is characteristic of exponential growth.
In this case, the number of bacteria \( N \) after \( x \) hours can be represented by the equation:
\[
N(x) = N_0 \cdot 2^x
\]
where:
- \( N_0 \) is the initial number of bacteria,
- \( x \) is the number of hours,
- \( 2^x \) represents the doubling effect after each hour.
Given that the initial population \( N_0 \) is 100, the specific equation for this situation would be:
\[
N(x) = 100 \cdot 2^x
\]
This function is exponential because the rate of growth (the number of bacteria) accelerates as time increases, with the population doubling every unit of time (in this case, every hour).
In this case, the number of bacteria \( N \) after \( x \) hours can be represented by the equation:
\[
N(x) = N_0 \cdot 2^x
\]
where:
- \( N_0 \) is the initial number of bacteria,
- \( x \) is the number of hours,
- \( 2^x \) represents the doubling effect after each hour.
Given that the initial population \( N_0 \) is 100, the specific equation for this situation would be:
\[
N(x) = 100 \cdot 2^x
\]
This function is exponential because the rate of growth (the number of bacteria) accelerates as time increases, with the population doubling every unit of time (in this case, every hour).