Question

A biologist places agar, a gel made from seaweed, in a Petri dish and infects it with bacteria. She uses the measurement of the growth ring to estimate the number of bacteria present. The biologist finds that the bacteria increase in population at an exponential rate of \(20 \%\) every 2 days. a) If the culture starts with a population of 5000 bacteria, what is the transformed exponential function in the form \(P=a(c)^{b x}\) that represents the population, \(P,\) of the bacteria over time, \(x,\) in days? b) Describe the parameters used to create the transformed exponential function. c) Graph the transformed function and use it to predict the bacteria population after 9 days.

Short answer

The transformed exponential function is \ P = 5000 \times 1.2^{0.5x} \ . After 9 days, the population is approximately 12442 bacteria.

Step-by-step solution

- Establish Base Parameters

The initial population of the bacteria, denoted as the initial value, is 5000. The growth rate is 20% every 2 days.

- Exponential Growth Function Form

The general form for exponential growth is given by the equation:\[ P = a \times c^{bx} \]Where: - \(P\) is the population at time \(x\)- \(a\) is the initial population- \(c\) is the growth factor- \(b\) is the frequency of the compounding period relative to the time unit.

- Determine the Growth Factor

Since the population increases by 20% every 2 days, the growth factor, \(c\), can be calculated as follows:\[ c = 1 + \frac{20}{100} = 1.2 \]

- Calculate the Frequency

The frequency \(b\) is the reciprocal of the time interval (in this case, every 2 days). Hence, \(b\) is calculated as follows:\[ b = \frac{1}{2} \]

- Combine Parameters to Form Function

Substituting the values of \(a\), \(c\), and \(b\) into the general form gives the transformed exponential function:\[ P = 5000 \times 1.2^{0.5x} \]

- Describe the Parameters

The parameters of the function are:- \(a = 5000\), which is the initial population of the bacteria.- \(c = 1.2\), which is the growth factor indicating a 20% increase every 2 days.- \(b = 0.5\), which adjusts for the growth rate in terms of days.

- Predict Population after 9 Days

To predict the population after 9 days, substitute \(x = 9\) into the transformed exponential function:\[ P = 5000 \times 1.2^{0.5 \times 9} \]Calculate the exponent first:\[ 0.5 \times 9 = 4.5 \]Then compute the population:\[ P = 5000 \times 1.2^{4.5} \approx 5000 \times 2.48832 \approx 12441.6 \]

- Graph the Transformed Function

Plot the exponential function on a graph where the x-axis represents time in days and the y-axis represents the population. The graph will be an upward-sloping curve reflecting the exponential growth of the bacteria.

Key concepts

exponential function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. This type of function models situations where quantities grow or decay at a constant rate relative to their current value. In general, an exponential function can be written as: \( y = a \times b^{x} \) where:

• \( y \) is the output value.
• \( x \) is the input or independent variable.
• \( a \) is the initial value or y-intercept.
• \( b \) is the base or growth factor.

In the context of population growth or decay, the exponential function captures how the population changes over time. Understanding its parameters (initial value and growth factor) is essential for predicting future values.

population growth

Population growth often follows an exponential trend, especially in uncontrolled environments like bacterial cultures. For these, the exponential function is used to estimate how a population increases over time due to reproduction. In our exercise, the biologist observes that the bacteria population grows by 20% every 2 days. This growth rate means the quantity of bacteria increases exponentially with time. Exponential growth is common in biological systems, but it can also apply to other scenarios such as viral infections or financial investments. Parameters like the initial population and the growth rate are key to accurately modeling and predicting population growth.

parameters in exponential functions

The parameters in an exponential function are crucial for constructing the model accurately. They include:

• Initial Value (\( a \)): This represents the starting amount in the modeled scenario. For the bacteria example, it is the initial population of 5000.
• Growth Factor (\( c \)): This is the base of the exponent and indicates how the quantity changes per time unit. Here, it is 1.2, reflecting a 20% increase every 2 days.
• Compounding Frequency (\( b \)): This adjusts the rate per the time unit, converting the growth interval to a daily rate. In our case, this frequency is 0.5 since the interval is 2 days.

These parameters combine in the function \( P = 5000 \times 1.2^{0.5x} \) to give a precise model of exponential bacterial growth. Properly understanding and calculating these parameters allow for accurate predictions.

graphing exponential functions

Graphing exponential functions provide a visual representation of how quantities grow or decay over time. Here are steps to graph the function \ \( P = 5000 \times 1.2^{0.5x} \):

1. Create a set of values for the independent variable \( x \), which represents time in days in this example.
2. Using the function, calculate corresponding \( y \) values, representing population at each time point.
3. Plot these \( (x, y) \) pairs on the graph with the x-axis for time and y-axis for population.
4. Draw a smooth curve through these points to illustrate the exponential growth. The graph will show an upward sloping curve because of the positive growth rate, clearly representing the exponential increase in the bacteria population over time. This visual aid aids comprehension and verifies the results from the function.