33. Under controlled conditions, the number of a certain type of bacteria doubles every 20 minutes. Suppose the initial number of the bacteria 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 (ln k)/(ln 2).
To determine how many 20-minute intervals it takes for the bacteria to reach a quantity of k, we can use the formula for exponential growth. The number of bacteria doubles every 20 minutes, leading to the expression (ln k)/(ln 2) to solve for the number of intervals.
A) k/2
This option incorrectly suggests a linear relationship between k and the number of intervals. The growth of bacteria is exponential, not linear, making this option not representative of the required intervals for the bacteria to reach a count of k.
B) k/20
Similar to option A, this choice implies a direct division of k by a constant value, which does not account for the exponential doubling behavior of the bacteria. Thus, it fails to accurately describe the time intervals needed for the bacteria to reach the count of k.
C) ln k
While the natural logarithm is a component of the correct formula, this option fails to include the necessary factor of ln 2, which accounts for the doubling time of the bacteria. Therefore, this expression does not represent the correct number of intervals.
D) (ln k)/(ln 2)
This option correctly represents the number of 20-minute intervals required for the number of bacteria to grow from 1 to k. The formula derives from the understanding that the population doubles every 20 minutes, making it essential to divide the natural logarithm of k by the natural logarithm of 2.
E) (ln k)/(ln 20)
This expression incorrectly uses 20 in the logarithm, which does not relate to the doubling time of the bacteria. The base of the logarithm must be 2 to accurately reflect the doubling behavior, thus making this option incorrect.
Conclusion
The correct answer, (ln k)/(ln 2), accurately reflects the exponential growth of the bacteria and the relationship between the number of intervals and the final quantity k. All other options fail to consider the nature of exponential growth or incorrectly apply logarithmic functions, leading to their inaccuracy in representing the situation.