Final answer:
The formula Pḿ = P₀ ⋅ 2⁴⁵⁶⁵(t/d) is used to calculate bacterial population growth over time. To find the population after 19 hours with a doubling time of 30 minutes, simply substitute the values into the formula and calculate, which typically involves large numbers best expressed in scientific notation.
Step-by-step explanation:
To find the population of bacteria in a culture after a given time using the exponential growth formula Pḿ = P₀ ⋅ 2⁴⁵⁶⁵(t/d), we need to have the initial population P₀, the time t in hours, and the doubling time d. If the doubling time for our bacteria is 30 minutes, which is 0.5 hours, and we want to know the population after 19 hours, we simply plug these numbers into the formula. For instance, let's say the initial population is 1,000 bacteria (P₀ = 1000).
To calculate Pḿ, the population after 19 hours, we use the formula:
Pḿ = 1000 ⋅ 2⁴⁵⁶⁵(19/0.5)
This calculation would involve determining how many doubling times are in 19 hours by dividing 19 by 0.5, which would result in 38 doubling times. Then, we calculate 2 to the power of 38:
2³⁸
Since this calculation is going to yield a very large number, it's often acceptable to use scientific notation or a calculator to find Pḿ. Remember that in a realistic biological context, factors like limited resources or environmental conditions would likely prevent such unchecked exponential growth.