Problem 8 - Algebra and Trigonometry

Question

Bacteria Culture It is observed that a certain bacteria culture has a relative growth rate of 12\(\%\) per hour, but in the 0presence of an antibiotic the relative growth rate is reduced to 5\(\%\) per hour. The initial number of bacteria in the culture is \(22 .\) Find the projected population after 24 hours for the following conditions. (a) No antibiotic is present, so the relative growth rate is 12\(\%\) . (b) An antibiotic is present in the culture, so the relative growth rate is reduced to \(5 \% .\)

Step-by-Step Solution

Answer

(a) 392 bacteria (b) 73 bacteria

Step 1: Understand the Problem

We need to calculate the future population of bacteria after 24 hours given two different growth rates: 12\(\%\) without antibiotic and 5\(\%\) with antibiotic. The initial population is 22 bacteria.

Step 2: Set Up the Formula

The formula to calculate the population after time \( t \) with continuous growth is \( P(t) = P_0 \cdot e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the relative growth rate, and \( t \) is the time in hours.

Step 3: Calculate Population Without Antibiotic

For part (a), without antibiotic, \( r = 0.12 \). Plug in the values: \( P_0 = 22 \), \( r = 0.12 \), \( t = 24 \). Thus, \( P(24) = 22 \cdot e^{0.12 \times 24} \).

Step 4: Solve for Part (a)

Calculate \( P(24) = 22 \cdot e^{2.88} \) which results in approximately \( 22 \cdot 17.84 \), so \( P(24) \approx 392.48 \). Thus, the population is approximately 392 bacteria.

Step 5: Calculate Population With Antibiotic

For part (b), with antibiotic, \( r = 0.05 \). Plug in the values: \( P_0 = 22 \), \( r = 0.05 \), \( t = 24 \). Thus, \( P(24) = 22 \cdot e^{0.05 \times 24} \).

Step 6: Solve for Part (b)

Calculate \( P(24) = 22 \cdot e^{1.2} \) which results in approximately \( 22 \cdot 3.32 \), so \( P(24) \approx 73.04 \). Thus, the population is approximately 73 bacteria.

Key Concepts

bacteria culture

Bacteria culture refers to the growth and multiplication of bacteria under controlled conditions. In a laboratory setting, scientists often observe these cultures to study growth patterns, test treatments, and develop antibiotics. Bacteria cultures can grow at different rates, influenced by factors like temperature, nutrients, and the presence of substances such as antibiotics. These microorganisms multiply through a process called binary fission, where one cell divides into two identical daughter cells. Understanding how bacteria cultures grow aids in fields such as medical research, food safety, and antibiotic development. In our exercise, a starting population of 22 bacteria gives us a baseline to calculate expected growth under different conditions.

relative growth rate

The relative growth rate of a bacteria culture is a measure that indicates how fast the bacteria population increases over time. It is often expressed as a percentage and represents the change in population per unit of time. A relative growth rate of 12\(\%\) per hour means the population grows by 12\(\%\) of its current size every hour. The equation used to calculate population growth is \( P(t) = P_0 \cdot e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the relative growth rate (in decimal form), and \( t \) is time in hours.

• High growth rates indicate rapid population increase, which could be due to optimal environmental conditions and absence of inhibitory agents such as antibiotics.
• Lower growth rates suggest slower population expansion, possibly due to limited resources or presence of inhibitory conditions.

population projection

Population projection refers to the estimation of future number of individuals in a population under specific conditions. In the context of bacteria culture, it involves calculating how large the population will be after a certain amount of time, given its initial size and growth rate. This projection can help researchers plan experiments and anticipate bacterial behaviors in varying environmental setups.In our example, we start with 22 bacteria, and using the formula \( P(t) = P_0 \cdot e^{rt} \), we project the population over 24 hours.

• Without antibiotic (12\(\%\) growth rate): The population grows significantly, reaching approximately 392 bacteria.
• With antibiotic (5\(\%\) growth rate): The growth rate is subdued, with a projection of roughly 73 bacteria.

Such projections are crucial for understanding and managing microbial growth in scientific and healthcare settings.

antibiotic effect

Antibiotics are substances that inhibit or stop bacteria growth. They play a vital role in controlling bacterial infections. However, not all bacteria react uniformly to antibiotics. Some may thrive slower, while others might be resistant altogether.In our exercise, the presence of an antibiotic reduces the growth rate from 12\(\%\) to 5\(\%\). This significant reduction showcases the effectiveness of the antibiotic in slowing down bacterial multiplication.

• Antibiotics target specific functions or structures within bacterial cells, hindering their ability to reproduce or survive.
• By reducing their growth rate, antibiotics minimize health risks associated with bacterial proliferation.
• Understanding how antibiotics affect bacteria is essential for developing effective treatments and preventing antibiotic resistance.

The reduced growth rate in our exercise highlights the profound impact antibiotics can have, emphasizing the importance of their prudent and strategic use in medicine.