26. Under controlled conditions, the number of a certain type of bacteria doubles every 20 minutes. Suppose the initial number of the bacteria in the laboratory is 1. Which of the following expressions represents the number of 20-minute intervals that it will take for the number of bacteria to equal k ?
Answer: D
The expression that represents the number of 20-minute intervals for the bacteria to equal k is (In(k))/(In(2)).
To determine the number of 20-minute intervals required for the bacterial population to reach k, we utilize the concept of exponential growth. Since the number of bacteria doubles every 20 minutes, the expression that accurately describes this relationship is (In(k))/(In(2)).
A) k/2
This option incorrectly suggests that the number of intervals is simply half of k. This does not account for the exponential growth of the bacteria, which doubles every 20 minutes rather than increasing linearly.
B) k/20
This choice implies a linear growth model, suggesting that the number of intervals is directly proportional to k divided by 20. Like option A, this fails to reflect the doubling nature of the bacterial growth occurring every 20 minutes.
C) In k
While this option involves the logarithm of k, it does not incorporate the necessary base that reflects the doubling rate of the bacteria. The growth pattern requires consideration of logarithms in base 2 to accurately represent how many intervals it takes to reach k.
D) (In(k))/(In(2))
This is the correct option as it accurately represents the number of 20-minute intervals needed for the bacteria to reach a population of k. By using logarithms, it accounts for the doubling nature of the population every 20 minutes, specifically indicating how many times the population would need to double to reach k.
E) (In(k))/(In(20))
This option incorrectly uses the logarithm of 20 as the base instead of 2. Since the bacteria double every 20 minutes, the growth is exponential with base 2, making this option invalid.
Conclusion
The correct expression, (In(k))/(In(2)), is the only one that accurately reflects the exponential growth of the bacterial population, accounting for the doubling every 20 minutes. All other options fail to consider the nature of exponential growth, leading to incorrect calculations regarding the number of intervals needed to reach k.