Problem 34 - Applied Calculus

Question

A colony of bacteria is of size \(S(t)=300 e^{0.1 t}\) after \(t\) hours. Find the average size during the first 12 hours (that is, from time 0 to time 12 ).

Step-by-Step Solution

Answer

The average size is approximately 580 bacteria.

Step 1: Understand the Problem and Set up the Integral

The size of the colony is given by the function \( S(t) = 300 e^{0.1t} \). We need to find the average size of the colony from time \( t = 0 \) to \( t = 12 \). The formula for average value of a function \( f(t) \) over the interval \([a, b]\) is given by: \[\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(t) \, dt\]Here, we'll use \( f(t) = 300 e^{0.1t} \), \( a = 0 \), and \( b = 12 \).

Step 2: Set up the Integral for Average Value

Using the average value formula, we set up the integral: \[\text{Average size} = \frac{1}{12 - 0} \int_{0}^{12} 300 e^{0.1t} \, dt\]This simplifies to:\[\frac{1}{12} \int_{0}^{12} 300 e^{0.1t} \, dt\]

Step 3: Solve the Integral

The integral to solve is \( \int 300 e^{0.1t} \, dt \). Using the integration rule for exponential functions, the antiderivative of \( e^{0.1t} \) is \( \frac{1}{0.1} e^{0.1t} = 10 e^{0.1t} \). Thus:\[\int 300 e^{0.1t} \, dt = 300 \times 10 e^{0.1t} = 3000 e^{0.1t}\]

Step 4: Evaluate the Definite Integral

Evaluate the definite integral from \( t = 0 \) to \( t = 12 \):\[3000 \left[e^{0.1t} \right]_{0}^{12} = 3000 (e^{1.2} - e^{0})\]Since \( e^{0} = 1 \), this becomes:\[3000 (e^{1.2} - 1)\]

Step 5: Calculate the Average Size

Substitute the evaluated definite integral back into the average formula:\[\frac{1}{12} \times 3000 (e^{1.2} - 1)\]This simplifies the expression to:\[250 (e^{1.2} - 1)\]

Step 6: Compute Numerical Value

Use a calculator to find \( e^{1.2} \approx 3.3201 \) and substitute it back:\[250 (3.3201 - 1) = 250 \times 2.3201 \approx 580.025\]Thus, the average size of the bacteria colony over the first 12 hours is approximately 580.

Key Concepts

Exponential Growth

Exponential growth refers to a process where the quantity increases at a rate proportional to its current value. In the context of the bacterial colony, the population grows continuously by a constant percentage, creating a curve that rises more steeply over time. In mathematical terms, this is represented by the formula for our bacteria colony:

• \(S(t) = 300 e^{0.1t}\)

Here, 300 is the initial size of the colony. The expression \(e^{0.1t}\) reflects the exponential growth factor, where \(e\) is the base of the natural logarithm. This function captures how the bacteria replicate continuously, increasing the population exponentially as each moment passes.
Exponential growth is common in nature and can be seen in various scenarios, such as population dynamics, finance, and nuclear chain reactions.

Definite Integral

Definite integrals are a major concept in calculus used to compute the accumulated quantity, like area under a curve between two specified limits. Imagine a curve plotted on a graph; the definite integral calculates the 'total' beneath this curve within a specific interval, providing meaningful insights into phenomenon like distances, areas, and averages.
The formula for the definite integral of a function \( f(t) \) from \(a\) to \(b\) is:

• \[ \int_{a}^{b} f(t) \, dt \]

In our case, the function \(f(t) = 300 e^{0.1t} \,\) is integrated from 0 to 12, which models the size of our bacterial colony over time.
This determines the area under the curve that represents the bacteria's growth, thereby providing the total sum of bacteria at various time points over the 12-hour period. Understanding definite integrals is crucial for solving real-world problems involving continuous change.

Bacteria Colony Modeling

Bacteria colony modeling uses mathematical equations and concepts to represent the growth dynamics of bacterial populations. These models help predict how bacteria grow over time, allowing biologists to study their behavior under different conditions.
The model in this exercise uses the function \(S(t) = 300 e^{0.1t}\), reflecting how many bacteria there are at any given time \(t\). An exponential model is particularly well-suited for bacteria because it captures their doubling behavior, assuming ideal conditions with unlimited resources.

• It provides insights into how quickly the colony can expand under favorable conditions.
• Appearing as a curve that steepens rapidly, exponential growth illustrates the rapid increase in biological systems.

This model can be used to project future population sizes and is crucial in areas such as medicine, ecology, and biotechnology, where controlling bacterial growth is important. Understanding these models helps in anticipating challenges related to bacterial infections and growth management.