Problem 74 - Precalculus Mathematics for Calculus

Question

A culture initially has 5000 bacteria, and its size increases by \(8 \%\) every hour. How many bacteria are present at the end of 5 hours? Find a formula for the number of bacteria present after \(n\) hours.

Step-by-Step Solution

Answer

After 5 hours, there are approximately 7346 bacteria. The formula is \(P(n) = 5000 \times (1.08)^n\).

Step 1: Understand the Problem

The culture starts with 5000 bacteria, and it grows by 8% every hour. We need to calculate how many bacteria are there after 5 hours and express the formula for any number of hours, \( n \).

Step 2: Calculate the Growth Factor

Since the bacteria culture grows by 8% every hour, the growth factor per hour is \(1 + 0.08 = 1.08\). This means each hour, the bacteria increase by a factor of 1.08.

Step 3: Determine the Formula for Bacteria Population

The formula to find the number of bacteria at any time \( n \) is based on exponential growth: \(P(n) = P_0 \times (1.08)^n\), where \(P_0 = 5000\) is the initial number of bacteria and \(n\) is the number of hours.

Step 4: Calculate Number of Bacteria After 5 Hours

Using the formula, substitute \(n = 5\): \[P(5) = 5000 \times (1.08)^5\]. This calculation determines the number of bacteria after 5 hours.

Step 5: Perform the Calculation

Calculate \((1.08)^5\), which is approximately 1.4693. Then multiply by the initial number of bacteria: \(5000 \times 1.4693 \approx 7346.5\). So, the number of bacteria after 5 hours is approximately 7346.

Key Concepts

Understanding Bacteria Culture

A bacteria culture refers to a population of bacterial cells growing in a controlled environment. This environment provides ideal conditions for growth, such as temperature, nutrients, and humidity. Bacteria reproduce at a remarkable rate, often doubling in number every few hours.
In our context, the exercise begins with a culture of 5000 bacteria. This starting point is important as it represents the initial size, denoted as \( P_0 \) in growth calculations.
As bacteria grow, observing the exponential increase is fascinating, which reflects in the rapid change in the number of cells. Understanding these growth patterns is crucial for microbiologists, who can predict how a culture will expand within a given period.

• Initial quantity: \( P_0 = 5000 \) bacteria
• Controlled environment: Ensures consistent growth rate
• Doubling behavior: Exhibits significant changes in short time spans

Significance of Growth Factor in Exponential Growth

The growth factor describes how much the culture increases over time. It's determined by the percentage increase in the population. Here, the bacteria culture grows by 8% every hour.
To calculate the growth factor, add the growth percentage to 1. Thus, for an 8% increase per hour, the growth factor will be \( 1 + 0.08 = 1.08 \). This value is pivotal because it influences how the initial population multiplies over time.
The growth factor enables us to construct formulas that can predict future population sizes.

• Growth percentage: 8% per hour
• Computed growth factor: 1.08
• Application: Used in exponential growth calculations

Exploring the Exponential Function for Population Growth

In mathematics, exponential growth is elegantly described using an exponential function. This involves raising a base number to a power, representing time progression. For bacteria growth, the formula \( P(n) = P_0 \times (1.08)^n \) captures this concept.
This function helps in finding how many bacteria will be present after \( n \) hours, taking into account the steady increase dictated by the growth factor.
By replacing \( n \) with specific values like 5 hours, we can find the population exact size at that time using \( P(5) = 5000 \times (1.08)^5 \).

• Formula: \( P(n) = P_0 \times (1.08)^n \)
• Prediction: Computes future bacterial population
• Practical use: Understanding dynamics of growth over time