Problem 12
Question
A colony of bacteria is growing exponentially. Initially, 500 bacteria were in the colony. The growth rate is \(20 \%\) per hour. (a) How many bacteria should be in the colony in 12 hours? (b) How many in 1 day? Use the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of bacteria.
Step-by-Step Solution
Verified Answer
Approximately 5512 bacteria after 12 hours; 60,755 bacteria after 1 day.
1Step 1: Identify Given Information
We are provided with initial data: - Initial population \( N_0 = 500 \))- Growth rate \( r = 20\% = 0.2 \))- Formula for exponential growth: \( N = N_0 e^{rt} \)
2Step 2: Calculate Bacteria Count at 12 Hours
To find the number of bacteria after 12 hours, plug the values into the formula:\[N = 500 \times e^{0.2 \times 12}\]Calculate:1. Multiply the rate by time: \( 0.2 \times 12 = 2.4 \)2. Calculate \( e^{2.4} \) using a calculator: approximately \( 11.023 \)3. Finally, multiply this with the initial population: \( 500 \times 11.023 = 5511.5 \)Thus, there are approximately 5512 bacteria after 12 hours.
3Step 3: Calculate Bacteria Count at 24 Hours
Let us determine how many bacteria will be present after 1 day (24 hours) using the formula:\[N = 500 \times e^{0.2 \times 24}\]Perform the calculations:1. Multiply the rate by time: \( 0.2 \times 24 = 4.8 \)2. Calculate \( e^{4.8} \) using a calculator: approximately \( 121.51 \)3. Multiply this with the initial population: \( 500 \times 121.51 = 60755 \)Therefore, there are approximately 60,755 bacteria after 24 hours.
Key Concepts
Understanding Bacteria GrowthDefining Initial PopulationInsights on Growth RateThe Role of Exponential Function
Understanding Bacteria Growth
Bacteria growth is an intriguing process studied extensively in biology and mathematics. In many cases, this growth occurs exponentially, meaning the population increases by a constant percentage over equal time periods.
Exponential growth is commonly observed in situations where resources are abundant, and limiting factors like space or nutrients are not a constraint.
This growth model can help to predict how fast a bacterial colony will expand under ideal conditions.
It's crucial to understand that in real-life situations, exponential growth doesn't continue indefinitely due to environmental constraints. However, for short periods and under controlled conditions, it provides a useful approximation of how quickly bacteria can multiply.
Exponential growth is commonly observed in situations where resources are abundant, and limiting factors like space or nutrients are not a constraint.
This growth model can help to predict how fast a bacterial colony will expand under ideal conditions.
It's crucial to understand that in real-life situations, exponential growth doesn't continue indefinitely due to environmental constraints. However, for short periods and under controlled conditions, it provides a useful approximation of how quickly bacteria can multiply.
Defining Initial Population
The initial population is the number of bacteria present at the very start of observation or analysis.
In our problem, the initial population is given as 500 bacteria. This number is denoted by the symbol \(N_0\) and serves as the starting point from which growth is calculated.
Understanding the initial population is crucial, as it anchors the growth process in a concrete starting value, allowing calculations of future populations.
When analyzing similar problems or conducting experiments, determining and knowing your initial population precisely ensures that your predictions and models remain accurate and reliable.
In our problem, the initial population is given as 500 bacteria. This number is denoted by the symbol \(N_0\) and serves as the starting point from which growth is calculated.
Understanding the initial population is crucial, as it anchors the growth process in a concrete starting value, allowing calculations of future populations.
When analyzing similar problems or conducting experiments, determining and knowing your initial population precisely ensures that your predictions and models remain accurate and reliable.
Insights on Growth Rate
The growth rate is a pivotal element when examining exponential growth. It refers to the percentage increase in the bacteria population after each time unit, usually expressed as a percentage.
In this particular example, the growth rate is 20% per hour, which we convert to a decimal as 0.2 for calculations.
This rate dictates how quickly the bacterial colony expands over time.
To apply this in calculations, multiply the growth rate by the time elapsed to see how much the exponent in the exponential formula will increase.
A higher growth rate means the population doubles more quickly, leading to faster and more dramatic increases in the population over time.
In this particular example, the growth rate is 20% per hour, which we convert to a decimal as 0.2 for calculations.
This rate dictates how quickly the bacterial colony expands over time.
To apply this in calculations, multiply the growth rate by the time elapsed to see how much the exponent in the exponential formula will increase.
A higher growth rate means the population doubles more quickly, leading to faster and more dramatic increases in the population over time.
The Role of Exponential Function
An exponential function is the mathematical representation of exponential growth.
The main formula to remember here is \(N = N_0 e^{rt}\), where \(N\) is the final number of bacteria, \(N_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time.
This formula utilizes the mathematical constant \(e\), roughly equal to 2.71828, which is the base of natural logarithms.
This constant is vital in calculations involving continuous growth or decay processes, such as bacterial growth.
When using the exponential function in calculations, a calculator or computer software is typically used to compute the value of \(e^{rt}\), as this is a crucial step for accurate predictions of the bacterial population at any given time.
The main formula to remember here is \(N = N_0 e^{rt}\), where \(N\) is the final number of bacteria, \(N_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time.
This formula utilizes the mathematical constant \(e\), roughly equal to 2.71828, which is the base of natural logarithms.
This constant is vital in calculations involving continuous growth or decay processes, such as bacterial growth.
When using the exponential function in calculations, a calculator or computer software is typically used to compute the value of \(e^{rt}\), as this is a crucial step for accurate predictions of the bacterial population at any given time.
Other exercises in this chapter
Problem 11
Approximate with a calculator. Round your answer to four decimal places. $$e^{-\pi}$$
View solution Problem 11
Write each logarithmic equation in its equivalent exponential form. $$-1=\ln \left(\frac{1}{e}\right)$$
View solution Problem 12
Solve the exponential equations exactly for \(x\). $$16^{x-1}=2^{x^{2}}$$
View solution Problem 12
Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$2^{\log _{2} 5}$$
View solution