Problem 35 - Algebra for College Students

Question

The number of bacteria present in a certain culture after \(t\) hours is given by the equation \(Q=Q_{0} e^{0.3 t}\), where \(Q_{0}\) represents the initial number of bacteria. If 6640 bacteria are present after 4 hours, how many bacteria were present initially? 2000

Step-by-Step Solution

Answer

The initial number of bacteria was 2000.

Step 1: Understand the Problem

We are given the equation \(Q = Q_{0} e^{0.3t}\), which describes the growth of bacteria over time. We need to find the initial number of bacteria \(Q_0\) given that after 4 hours, \(Q = 6640\).

Step 2: Set Up the Equation

Substitute the given values into the equation \(Q = Q_{0} e^{0.3t}\). We know \(Q = 6640\) after \(t = 4\) hours: \(6640 = Q_{0} e^{0.3 imes 4}\).

Step 3: Calculate the Exponent

Simplify the exponent by calculating: \(0.3 \times 4 = 1.2\). So the equation becomes \(6640 = Q_{0} e^{1.2}\).

Step 4: Isolate the Initial Quantity

Isolate \(Q_{0}\) by dividing both sides of the equation by \(e^{1.2}\): \(Q_{0} = \frac{6640}{e^{1.2}}\).

Step 5: Compute the Exponent

Use a calculator to find the value of \(e^{1.2}\). Approximating, \(e^{1.2} \approx 3.3201\).

Step 6: Calculate the Initial Number of Bacteria

Divide 6640 by 3.3201 to find \(Q_{0}\): \(Q_{0} = \frac{6640}{3.3201} \approx 2000\). Thus, the initial number of bacteria was approximately 2000.

Key Concepts

Bacterial Growth

Bacterial growth is a classic example of exponential growth in biology. In this process, the number of bacteria increases rapidly over time. Bacteria reproduce by splitting into two, and if the conditions are ideal, this can happen very quickly.
In mathematical terms, the growth is often represented by an equation:

• \(Q(t) = Q_0 e^{rt}\)

Here, \(Q(t)\) is the number of bacteria at time \(t\), \(Q_0\) is the initial number of bacteria, \(e\) is the base of the natural logarithm, and \(r\) is the growth rate.
This formula shows that the quantity of bacteria increases exponentially as time progresses. It's important to understand this behavior because it helps in predicting how fast a bacterial population can grow in a given period when conditions permit.

Initial Quantity Calculation

Calculating the initial number of bacteria is a reverse engineering process of exponential equations. This often requires some understanding of rearranging formulas and using logarithmic functions.
To find the initial amount \(Q_0\) when given the population after a certain time, the exponential growth equation \(Q = Q_0 e^{rt}\) needs to be rearranged to isolate \(Q_0\):

• \(Q_0 = \frac{Q}{e^{rt}}\)

Using this rearrangement, given a future population number, you can calculate how many bacteria you started with. For example, given that the population is 6640 after 4 hours with a growth rate of 0.3, you would calculate the exponent and then divide the final population by these factors to find \(Q_0\).
This is a practical application in various scientific fields such as microbiology and epidemiology, providing critical information about how quickly and extensively bacteria and other organisms can proliferate.

Solving Exponential Equations

Solving exponential equations usually involves several steps to isolate the variable of interest, which may lead you to use logarithms at times.
For instance, with an equation like \(6640 = Q_0 e^{1.2}\), you first need to isolate \(Q_0\). This involves dividing both sides by \(e^{1.2}\) to solve for \(Q_0\):

• \(Q_0 = \frac{6640}{e^{1.2}}\)

Now, the challenge is computing \(e^{1.2}\). This requires a calculator that can handle exponentials and once you determine \(e^{1.2}\), you will perform the division.Understanding each step and being comfortable with calculator functions is essential to solve these equations accurately. In essence, practicing these calculations will improve your ability to correctly determine initial quantities or predict future values in exponential growth scenarios.