Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Growth of bacteria A certain culture initially contains 10,000 bacteria and increases by \(20 \%\) every hour. (a) Find a formula for the number \(N(t)\) of bacteria present after \(t\) hours. (b) How many bacteria are in the culture at the end of 10 hours?

Step-by-Step Solution

Answer

(a) \(N(t) = 10,000 \times (1.2)^t\); (b) Approximately 61,917 bacteria.

Step 1: Understanding Exponential Growth

Since the bacteria increase by 20% every hour, they grow exponentially. The general formula for exponential growth is given by \( N(t) = N_0 \times (1 + r)^t \), where \( N_0 \) is the initial amount, \( r \) is the rate of increase, and \( t \) is the time in hours.

Step 2: Plug in the Known Values

The initial number of bacteria \( N_0 \) is 10,000, and the rate of increase \( r \) is 0.2 (20%). Substitute these values into the formula: \[ N(t) = 10,000 \times (1 + 0.2)^t \].

Step 3: Simplify the Formula

Simplify the formula by calculating \(1 + 0.2\): \[ N(t) = 10,000 \times (1.2)^t \]. Here, \(1.2\) represents the growth factor every hour.

Step 4: Calculate the Number of Bacteria after 10 Hours

We need to find \(N(10)\), which is the number of bacteria after 10 hours. Substitute \(t = 10\) into the simplified formula: \[ N(10) = 10,000 \times (1.2)^{10} \].

Step 5: Compute \((1.2)^{10}\)

Calculate \((1.2)^{10}\) using a calculator: \((1.2)^{10} \approx 6.1917364224\).

Step 6: Find the Number of Bacteria

Multiply 10,000 by 6.1917364224 to find \(N(10)\): \[ N(10) = 10,000 \times 6.1917364224 \approx 61,917.364224 \]. Therefore, after rounding, there are approximately 61,917 bacteria.

Key Concepts

Bacterial Culture

A bacterial culture refers to a population of bacteria growing in a given environment, such as a lab dish or a defined medium. These cultures are often used in experiments to understand bacterial behaviors and growth patterns. In our exercise, the initial bacterial culture starts with 10,000 bacteria. The environment is set in such a way that the bacteria thrive and multiply efficiently.
To control and observe such growth, scientists often use defined parameters, like temperature, nutrients, and oxygen levels.

• The initial amount of bacteria is crucial as it sets the starting point for calculations.
• The culture is monitored to observe growth over time, helping us understand bacterial behavior.

Understanding bacterial cultures is fundamental in microbiology, as it helps devise strategies to manage bacterial growth in various fields, from medicine to agriculture.

Growth Rate

The growth rate in a bacterial culture context refers to how quickly the bacteria population increases over time. It is often expressed as a percentage, indicating how much the population grows in a unit of time. In our scenario, the bacteria increase by 20% every hour, meaning every hour the new amount includes 20% more bacteria than the hour before.
Understanding the growth rate helps in predicting the bacterial culture's future size, assisting in resource planning and management. For a 20% hourly growth:

• If you start with 10,000 bacteria, after one hour, there will be 10,000 plus an additional 20% of that, making it 12,000 bacteria.
• This consistent percentage increase per hour results in exponential growth, meaning the size of the bacteria population becomes increasingly larger with time.

Recognizing and calculating growth rates is essential for studies in microbiology and biotechnology.

Exponential Function

Exponential functions are mathematical expressions that model situations where growth or decay accelerates over time. In the context of bacterial culture, an exponential function describes how the population multiplies at a constant percentage rate over intervals of time.
Exponential growth is represented with the formula:

• \( N(t) = N_0 \times (1 + r)^t \)
• \( N(t) \) represents the number of bacteria at time \( t \)
• \( N_0 \) is the initial number of bacteria
• \( r \) is the growth rate expressed as a decimal
• \( t \) is the time variable

The exponential function is crucial because it provides a concise formula to predict future outputs under constant growth rates. This has practical applications in fields like ecology, epidemiology, and economics.

Algebraic Formula

Algebraic formulas are essential tools that provide solutions to various scientific and mathematical problems. In the case of bacterial growth, the algebraic formula derived from the exponential growth function helps calculate the future amount of bacteria.
The specific formula used in this exercise is:

• \( N(t) = 10,000 \times (1.2)^t \)
• The number 10,000 is the initial count of bacteria.
• The growth factor of 1.2 accounts for the 20% increase each hour.

Understanding and setting up algebraic formulas allow us to predict outcomes by manipulating and substituting the known variables. Such calculations are vital in scientific fields for making predictions and analyzing data trends.