Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Bacteria Growth The number \(N\) of bacteria in a culture is given by the model \(N=100 e^{k t}\), where \(t\) is the time (in hours), with \(t=0\) corresponding to the time when \(N=100\). When \(t=6\), there are 140 bacteria. How long does it take the bacteria population to double in size? To triple in size?

Step-by-Step Solution

Answer

It takes approximately \(t = ln(2)/k \) hours for the bacteria population to double, and \(t = ln(3)/k \) hours for the population to triple. Actual values requires calculating \(k = ln(1.4)/6\) and substituting the result.

Step 1: Solve for the constant \(k\)

When \(t=6\), \(N=140\). Substituting these values into the formula: \[140 = 100 e^{k * 6}\] To isolate \(k\), we could use the properties of natural logarithms. First divide both sides by 100 i.e., \[1.4 = e^{6k}\], then take the natural logarithm of both sides: \[ln(1.4) = 6k\]. Solving for \(k\) gives: \[k = ln(1.4)/6\]

Step 2: Find the time for the population to double

Now that we have \(k\), we can determine how long it takes the population to double. We want to find \(t\) when \(N=200\), because 200 is double the original population. Substituting \(N=200\) and \(k = ln(1.4)/6 \) into the formula gives us: \[2 = e^{k*t}\], then take the natural logarithm of both sides: \[ln(2) = k*t\], solving for \(t\) gives: \[t = ln(2)/k \]

Step 3: Find the time for the population to triple

We now need to find \(t\) when \(N=300\), as 300 is triple the original population. Substituting these values into the formula gives: \[3 = e^{k*t}\], then take the natural logarithm of both sides: \[ln(3) = k*t\], solving for \(t\) gives: \[t = ln(3)/k \]

Key Concepts

Bacteria Growth

Bacteria growth is a fascinating example of exponential growth found in nature. In simple terms, exponential growth occurs when the rate of increase of a quantity is proportional to its current value. For bacteria, this means that as the population increases, it will grow even faster over time. Scientists often use mathematical models to describe this type of growth, with the formula \(N = N_0 e^{kt}\) explaining how the number of bacteria \(N\) changes over time \(t\).
\(N_0\) is the initial number of bacteria, \(e\) is the base of the natural logarithm (approximately 2.718), and \(k\) is the rate of growth constant. Such models are used to predict how bacterial cultures will evolve over different timescales, which helps in many scientific and medical applications, such as understanding infections and designing antibiotics.
The ability of bacteria to multiply quickly under optimal conditions can lead to very large numbers in a short period of time, illustrating the power of exponential growth in biological systems.

Natural Logarithms

Natural logarithms, often represented by \(ln\), are a mathematical concept that simplifies calculations involving exponential growth and decay. They are the logarithms to the base \(e\), a special mathematical constant roughly equal to 2.718. In the context of bacteria growth, natural logarithms help solve equations of the form \(N = N_0 e^{kt}\) for unknown quantities like time \(t\) or the growth rate \(k\).
Natural logarithms are extremely useful when trying to isolate or solve for variables in exponential equations, as they transform the exponentiation into multiplication, which is much easier to handle in algebraic expressions.

• Example: If you have \(e^{kt} = 1.4\), taking the natural logarithm will give you \(kt = ln(1.4)\).
• Another key property: the natural logarithm of 1 is 0, which can be helpful in simplifying some calculations.

Understanding natural logarithms is essential for grasping how exponential equations are manipulated and solved in scientific contexts.

Doubling Time

Doubling time is a specific calculation used in exponential growth that tells us how long it will take for a quantity to double in size. In bacteria growth, doubling time helps predict how quickly a population will expand under certain conditions. To find the doubling time, we use the equation \(t_d = \frac{ln(2)}{k}\), where \(t_d\) is the doubling time and \(k\) is the rate of growth constant.
Because exponential growth can be quite rapid, knowing the doubling time can be critical for anticipating the behaviors of biological systems.

• The natural logarithm of 2, \(ln(2)\), is approximately 0.693.
• This formula allows you to predict how fast your bacterial culture will reach a certain size, which is particularly important in areas like microbiology or environmental science.

The shorter the doubling time, the faster the bacteria population is increasing, which could have significant implications depending on the context.

Rate of Growth

The rate of growth in the context of bacteria is a measure that describes how fast the bacteria population increases. This is represented by the constant \(k\) in the exponential growth model \(N = N_0 e^{kt}\). Calculating \(k\) is key to understanding the dynamics of bacterial growth.
In practice, to find \(k\), you need data on the population at two different times. For instance, if you know how the population changes from \(N_0\) to \(N\) over a time period \(t\), you can determine \(k\) using the formula \(k = \frac{ln \left(\frac{N}{N_0}\right)}{t}\).

• This constant helps to standardize growth measurements, allowing direct comparisons between different bacterial cultures or conditions.
• The rate of growth informs us about the environmental factors or conditions that might affect bacterial reproduction, such as temperature, nutrient availability, or the presence of toxins.

Understanding the rate of growth provides insights into how quickly a bacterial culture can expand or how effective certain treatments might be in controlling microbial populations.