Problem 48 - Applied Calculus

Question

A culture of bacteria is growing at the rate of \(20 e^{0.8 t}\) cells per day, where \(t\) is the number of days since the culture was started. Suppose that the culture began with 50 cells. a. Find a formula for the total number of cells in the culture after \(t\) days. b. If the culture is to be stopped when the population reaches 500 , when will this occur?

Step-by-Step Solution

Answer

a. The formula for total cells after \(t\) days is \(P(t) = 25e^{0.8t} + 25\). b. The culture reaches 500 cells in approximately 3.68 days.

Step 1: Understanding the Growth Rate

We are given that the rate of growth of the bacteria culture is \(20e^{0.8t}\). This represents the derivative of the total population with respect to time, \(\frac{dP}{dt}\), where \(P\) is the population of the bacteria at time \(t\).

Step 2: Setting Up the Differential Equation

The rate of growth given by \(\frac{dP}{dt} = 20e^{0.8t}\) is a differential equation that we need to solve to find the population function \(P(t)\).

Step 3: Solving the Differential Equation

To find \(P(t)\), integrate \(\frac{dP}{dt} = 20e^{0.8t}\) with respect to \(t\). This gives:\[ P(t) = \int 20e^{0.8t} \, dt = \frac{20}{0.8} e^{0.8t} + C = 25e^{0.8t} + C \]

Step 4: Using Initial Conditions

We know the culture started with 50 cells, i.e., \(P(0) = 50\). Substitute \(t = 0\) and \(P(0) = 50\) into the equation to solve for \(C\):\[ 50 = 25e^{0} + C \]\[ C = 50 - 25 = 25 \]

Step 5: Writing the General Formula

We now have the total population formula:\[ P(t) = 25e^{0.8t} + 25 \]

Step 6: Solving for When Population Reaches 500

Set the population equation equal to 500 and solve for \(t\):\[ 25e^{0.8t} + 25 = 500 \]\[ 25e^{0.8t} = 475 \]\[ e^{0.8t} = 19 \]Take the natural logarithm of both sides:\[ 0.8t = \ln(19) \]\[ t = \frac{\ln(19)}{0.8} \]

Step 7: Calculating the Time

Calculate \(t\) using a calculator, \(t \approx \frac{2.944}{0.8} \approx 3.68\) days.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to describe how things change over time.
They involve derivatives, which represent the rate of change.
In our exercise, the differential equation is given as \( \frac{dP}{dt} = 20e^{0.8t} \). This equation tells us how fast the bacteria population is growing.To solve a differential equation means to find a function that satisfies it.
Here, we are looking for the population function \( P(t) \) that tells us the total number of bacteria at any given time \( t \).
Integration is the process we use to solve the differential equation by reversing differentiation to find \( P(t) \) from \( \frac{dP}{dt} \).The initial condition, like knowing the culture started with 50 cells, helps us calculate constants after integration. This assures our solution fits the specific scenario we are examining.

Population Dynamics

Population dynamics explore how populations change based on various factors.
The growth rate in our exercise, \( 20e^{0.8t} \), indicates that the population is increasing exponentially with time.
This model represents unrestrained growth, often seen in bacteria under ideal conditions.With an exponential model, the population grows faster as time progresses.
This is typical in environments without resource limits, showing how initially small populations can quickly become large.
As with our exercise, when a population reaches a specific target, like 500 cells, the dynamics can shift to new models or involve further interventions.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios through mathematical expressions.
Models, like our exponential growth model, help predict future behavior based on current data.
In bacterial growth scenarios, a simple mathematical model can illustrate complex biological processes efficiently.
Our formula for population, \( P(t) = 25e^{0.8t} + 25 \), is such a model.
It uses mathematical symbols and functions to represent the growth of bacteria over time.Mathematical models can be refined by adding more factors or variables if additional data is available.
Initially, they provide a snapshot of expected behavior under given assumptions.
These models are vital in areas like biology, economics, and engineering, where they guide decisions and plan outcomes.