Problem 40 - Intermediate Algebra

Question

The number of bacteria present at a given time under certain conditions is given by the equation \(Q=5000 e^{0.05 t}\), where \(t\) is expressed in minutes. How many bacteria are present at the end of 10 minutes? 30 minutes? 1 hour?

Step-by-Step Solution

Answer

8258 at 10 mins, 12214 at 30 mins, 100949 at 1 hour.

Step 1: Substitute and Calculate for 10 Minutes

To find the number of bacteria at the end of 10 minutes, substitute \(t = 10\) into the equation \(Q = 5000 e^{0.05t}\). This gives us \(Q = 5000 e^{0.05 imes 10}\). Simplifying the exponent, we find \(Q = 5000 e^{0.5}\). Calculate the value of \(e^{0.5}\) and multiply by 5000 to find the answer.

Step 2: Substitute and Calculate for 30 Minutes

To find the number of bacteria at the end of 30 minutes, substitute \(t = 30\) into the equation. This gives us \(Q = 5000 e^{0.05 imes 30}\). Simplifying the exponent, we find \(Q = 5000 e^{1.5}\). Calculate the value of \(e^{1.5}\) and multiply by 5000 to determine the number of bacteria.

Step 3: Convert to Hours and Calculate

To find the number of bacteria at the end of 1 hour, note that 1 hour is equivalent to 60 minutes. Substitute \(t = 60\) into the equation: \(Q = 5000 e^{0.05 imes 60}\). Simplify the exponent to get \(Q = 5000 e^{3}\). Calculate the value of \(e^{3}\) and multiply by 5000 to get the final result.

Key Concepts

Bacteria Population

Bacteria are tiny organisms that can multiply rapidly under the right conditions. Imagine starting with 5000 bacteria in a container. If these bacteria grow exponentially, their numbers can increase very quickly. The growth is often governed by certain conditions, like temperature and nutrients, which are reflected in a mathematical model. The equation we use here is called an exponential function, showing how the population expands over time. The formula used in the exercise is: \[Q = 5000 e^{0.05t}\] where:

• \(Q\) represents the quantity of bacteria at a given time \(t\)
• The initial number of bacteria is 5000
• \(e\) is a constant approximately equal to 2.71828

This formula helps predict the population without needing to count each bacterium individually.

Exponential Function

An exponential function describes a process that increases rapidly at a constant rate. It's defined mathematically as a function of the form \(f(x) = a \, e^{bx}\), where \(a\) and \(b\) are constants, and \(e\) is the base of the natural logarithm. In biology, it's often used to model growth, such as populations of bacteria, since these are capable of doubling at regular intervals. For the bacteria population:

• \(a = 5000\), the initial count of bacteria
• \(b = 0.05\), representing the growth rate per time unit

This means the population grows continually by 5% every minute, demonstrating how quickly exponential growth can compound. Solving such equations involves simplifying the exponent and calculating the resulting number using the constant \(e\). In practice, this requires use of a calculator to compute power of \(e\).

Time Conversion

Time conversion is a critical skill in solving problems involving rates of change over time. Typically, problems define time in terms fitting the context of the growth. In this exercise, time \(t\) is initially given in minutes. However, when asked to find how many bacteria are present at the end of 1 hour, recognizing that 1 hour equals 60 minutes is essential. Steps to convert time:

• Recognize that 1 hour equals 60 minutes
• Substitute 60 for \(t\) when calculating for 1 hour

Such conversion ensures you calculate with the correct time duration. After converting, apply this time in the exponential growth formula to find the new population size. This conversion maintains the relationship between time units and the growth rate.