Problem 71
Question
A bacteria population is 4000 at time \( t = 0 \) and its rate of growth is \( 1000 \cdot 2^t \) bacteria per hour after \( t \) hours. What is the population after one hour?
Step-by-Step Solution
Verified Answer
The population after one hour is 6000 bacteria.
1Step 1: Understand the Problem
We need to find out the bacteria population after one hour, given an initial population and a rate of growth formula. The initial population at time \( t = 0 \) is 4000, and the growth rate is \( 1000 \cdot 2^t \) bacteria per hour.
2Step 2: Identify Given Values
At \( t = 0 \), the initial population \( P_0 \) is 4000. The growth rate of the bacteria is calculated with the formula \( 1000 \cdot 2^t \). We want to find the population after \( t = 1 \) hour.
3Step 3: Calculate Growth After One Hour
Using the growth rate formula \( 1000 \cdot 2^t \), plug in \( t = 1 \):\[ 1000 \cdot 2^1 = 1000 \cdot 2 = 2000 \]So, 2000 bacteria are added after one hour.
4Step 4: Determine the Total Population
Add the bacteria grown during the first hour to the initial population:\[ \text{Total Population} = 4000 + 2000 = 6000 \]Hence, the population after one hour is 6000.
Key Concepts
Initial PopulationExponential Growth FormulaRate of Growth
Initial Population
In order to understand bacterial population dynamics, it's essential to grasp the concept of the initial population. This term refers to the number of bacteria present at the beginning of an observation period. It's the starting point for any calculations related to growth or decline of a population.
For instance, consider when we begin with 4000 bacteria. This figure is crucial because all future growth estimates stem from this initial count. It establishes the baseline against which the rate of growth is applied.
An initial population helps in setting up equations for population growth. Knowing this number allows us to apply mathematical models to predict changes over time.
For instance, consider when we begin with 4000 bacteria. This figure is crucial because all future growth estimates stem from this initial count. It establishes the baseline against which the rate of growth is applied.
An initial population helps in setting up equations for population growth. Knowing this number allows us to apply mathematical models to predict changes over time.
Exponential Growth Formula
Exponential growth is a key concept in understanding how populations such as bacteria increase over time. The exponential growth formula models situations where a population grows at a rate proportional to its current size. This can be expressed mathematically as:\[ N(t) = N_0 \cdot e^{rt} \]Where:
Understanding exponential growth is vital for predicting how rapidly a population can increase under optimal conditions.
- \( N(t) \) is the population size at time \( t \).
- \( N_0 \) is the initial population size.
- \( r \) is the growth rate constant.
- \( t \) is time.
Understanding exponential growth is vital for predicting how rapidly a population can increase under optimal conditions.
Rate of Growth
The rate of growth of a population describes how fast or slow that population changes over time. It's an essential component in predicting future population sizes. In mathematical models, the rate of growth tells us how the population size increases relative to the existing population.
In our example, the growth rate is given by the formula \( 1000 \cdot 2^t \). This indicates that for every hour \( t \), the population grows by \( 1000 \) times \( 2^t \). This exponential factor means the population doubles with each passing hour.
By understanding the rate of growth, we can calculate the number of bacteria added each hour. For instance, at \( t = 1 \), the calculation is \( 1000 \cdot 2^1 = 2000 \) new bacteria. Hence, knowing the rate of growth helps us in making future projections accurately and understanding the population dynamics in biological systems.
In our example, the growth rate is given by the formula \( 1000 \cdot 2^t \). This indicates that for every hour \( t \), the population grows by \( 1000 \) times \( 2^t \). This exponential factor means the population doubles with each passing hour.
By understanding the rate of growth, we can calculate the number of bacteria added each hour. For instance, at \( t = 1 \), the calculation is \( 1000 \cdot 2^1 = 2000 \) new bacteria. Hence, knowing the rate of growth helps us in making future projections accurately and understanding the population dynamics in biological systems.
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