Problem 66 - Algebra and Trigonometry

Question

Bacteria Culture 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

There are about 7347 bacteria after 5 hours. The formula is \( N(n) = 5000 \times (1.08)^n \).

Step 1: Understand the Problem

We have a bacteria culture initially with 5000 bacteria, and the number of bacteria increases by 8% every hour. We need to determine how many bacteria are present after 5 hours and also find a general formula for any n hours.

Step 2: Determine the Growth Rate

The growth rate of bacteria is given as 8% per hour. In mathematical terms, this means the bacteria count grows by a factor of 1 + 0.08 = 1.08 each hour.

Step 3: Apply the Growth Formula for Hours

The formula for exponential growth is given by: \[ N(t) = N_0 imes (1 + r)^t \]where \(N(t)\) is the number of bacteria at time \(t\), \(N_0\) is the initial number of bacteria, \(r\) is the growth rate, and \(t\) is the time in hours. Given \(N_0 = 5000\), \(r = 0.08\), and for \(t = n\):\[ N(n) = 5000 imes (1.08)^n \]

Step 4: Calculate the Number of Bacteria After 5 Hours

Substitute \(n = 5\) into the formula:\[ N(5) = 5000 imes (1.08)^5 \]Calculate \(1.08^5\):\[ 1.08^5 \approx 1.4693 \]Now multiply by 5000:\[ N(5) \approx 5000 imes 1.4693 = 7346.5 \]Therefore, rounding to a whole number, there are approximately 7347 bacteria after 5 hours.

Key Concepts

Bacteria Culture

Bacteria cultures are populations of bacteria that are grown under controlled conditions, usually in laboratory settings. Scientists often study bacteria cultures to understand the biological processes that occur within these microorganisms.

One notable trait of bacteria is their ability to replicate quickly, making them ideal candidates for studies on growth patterns.

A typical bacteria culture starts with a certain number of bacteria, known as the initial bacteria count. Understanding this concept is crucial because the initial count forms the basis for predicting the future size of the culture using mathematical models.

Growth Rate

The growth rate of a bacteria culture is defined as the percentage increase in population size over a certain time period.

In our exercise, the bacteria grow at a rate of 8% per hour. This means that each hour, the original number of bacteria increases by 8% more than their previous total.

• A growth rate expressed as a percentage is common in many fields.
• It’s essential to convert this percentage into a decimal for calculation purposes.
• For an 8% growth rate, the conversion is done by dividing by 100, resulting in 0.08.

Understanding growth rates allows scientists and students to calculate changes in bacterial populations accurately, predict future sizes, and facilitate comparisons across different scenarios.

Exponential Growth Formula

Exponential growth describes a process where the rate of change of a quantity is proportional to the current amount of that quantity.

In the case of bacteria culture, exponential growth can be modeled using the exponential growth formula: \[ N(t) = N_0 \times (1 + r)^t \]where:

• \(N(t)\) represents the total 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 in hours.

For our problem scenario, substituting the given values (initial size = 5000, growth rate = 8% or 0.08) into the formula provides a generalized model: \(N(n) = 5000 \times (1.08)^n \). This model predicts the population for any number of hours \(n\), such as 5 hours in our example, yielding approximately 7347 bacteria. This formula thus serves as a potent tool in predicting long-term trends in biological and other scientific studies.