Problem 29
Question
Suppose that the bacteria in a colony can grow unchecked, by the law of exponential change. The colony starts with 1 bacterium and doubles every half- hour. How many bacteria will the colony contain at the end of 24 hours? (Under favorable laboratory conditions, the number of cholera bacteria can double every 30 min. In an infected person, many bacteria are destroyed, but this example helps explain why a person who feels well in the morning may be dangerously ill by evening.)
Step-by-Step Solution
Verified Answer
The colony will contain 281,474,976,710,656 bacteria at the end of 24 hours.
1Step 1: Determine the Growth Formula
The growth of a population according to the law of exponential change is given by the formula \( P(t) = P_0 \times 2^{t/T} \). Here, \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, and \( T \) is the doubling time.
2Step 2: Identify the Known Values
The problem states that \( P_0 = 1 \) (starting with one bacterium), \( t = 24 \) hours, and the doubling time \( T = 0.5 \) hours, as the population doubles every half hour.
3Step 3: Convert Time to Doubling Periods
To find the number of doubling periods in 24 hours, divide the total time by the doubling time: \( N = \frac{24}{0.5} = 48 \). So, there are 48 half-hour doubling periods in 24 hours.
4Step 4: Calculate the Population
Substitute the values into the formula: \( P(24) = 1 \times 2^{48} \). Compute \( 2^{48} \), which gives \( 281,474,976,710,656 \).
5Step 5: Interpret the Result
After 24 hours, the number of bacteria in the colony is \( 281,474,976,710,656 \), assuming no environmental limiting factors.
Key Concepts
Bacterial GrowthPopulation DoublingExponential ChangeGrowth Formula
Bacterial Growth
Bacterial growth is a fascinating example of exponential growth in biology. It refers to how bacteria multiply rapidly under ideal conditions. One bacterium can split into two, these two can become four, and so on. This makes bacteria unique because their populations can increase very fast in a very short time.
The exercise focused on a single bacterium growing in a laboratory setting. Here, it was assumed no factors were limiting their growth, like nutrient shortage or competition. When you start with one bacterium and it grows unchecked, it follows a pattern where the numbers grow exponentially because the population size doubles at regular intervals. This is why bacterial infections can become severe so quickly if left unchecked, much like the morning-to-evening illness example.)
The exercise focused on a single bacterium growing in a laboratory setting. Here, it was assumed no factors were limiting their growth, like nutrient shortage or competition. When you start with one bacterium and it grows unchecked, it follows a pattern where the numbers grow exponentially because the population size doubles at regular intervals. This is why bacterial infections can become severe so quickly if left unchecked, much like the morning-to-evening illness example.)
Population Doubling
Population doubling is when a population doubles in size. In the context of bacterial growth, this doubling happens during a consistent time interval. For the described bacteria colony, the population doubles every 30 minutes.
Understanding how long it takes for a population to double is essential for predicting how large a population will grow over time. It also helps us understand more about control methods necessary for managing populations, especially in disease control.
Understanding how long it takes for a population to double is essential for predicting how large a population will grow over time. It also helps us understand more about control methods necessary for managing populations, especially in disease control.
- Double in size = twice as many.
- The doubling time here was given as 0.5 hours or 30 minutes.
Exponential Change
Exponential change means that as time progresses, the quantity grows by multiplying itself by a constant factor. Such growth is different from linear growth, where quantities increase by the same fixed amount each step.
Bacteria exhibit exponential change because they multiply by dividing in two repeatedly. With each division, the total number becomes much larger quite quickly. This type of change can be portrayed by a mathematical formula, allowing us to predict future population sizes. In this case:
Bacteria exhibit exponential change because they multiply by dividing in two repeatedly. With each division, the total number becomes much larger quite quickly. This type of change can be portrayed by a mathematical formula, allowing us to predict future population sizes. In this case:
- Start with one bacterium, and it keeps doubling.
- As time goes on, the number of bacteria grows more and more rapidly.
Growth Formula
The growth formula is a mathematical way to express how quantities increase over time under exponential growth conditions. The formula used in this exercise is:
\[ P(t) = P_0 imes 2^{t/T} \]
Where:
\[ P(t) = P_0 imes 2^{t/T} \]
Where:
- \( P(t) \) = population at time \( t \)
- \( P_0 \) = initial population (here, 1 bacterium)
- \( T \) = time for the population to double (0.5 hours in this case)
Other exercises in this chapter
Problem 28
Gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^
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Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\frac{1}{2} \ln \frac{1+x}{1-x}$$
View solution Problem 29
Find the derivative of \(y\) with respect to the appropriate variable. $$y=(1-t) \operatorname{coth}^{-1} \sqrt{t}$$
View solution Problem 29
Use l'Hôpital's rule to find the limits.. $$\lim _{x \rightarrow 0} \frac{x 2^{x}}{2^{x}-1}$$
View solution