Problem 73 - Precalculus : Building Concepts and Connections

Question

Suppose the population of a colony of bacteria doubles in 12 hours from an initial population of 1 million. Find the growth constant \(k\) if the population is modeled by the function \(P(t)=P_{0} e^{k t} .\) When will the population reach 4 million? 8 million?

Step-by-Step Solution

Answer

The growth constant \(k\) is \(ln(2) / 12\). To find out when the population would reach 4 million and 8 million, replace \(P(t)\) with 4 million and 8 million respectively in the exponential growth equation and solve for \(t\).

Step 1: Determining the growth constant k

The population of the bacteria colony doubles in 12 hours. Therefore, at t=12, we can say that \(P(t) = 2 * P_0\), where \(P_0\) is the initial population of the colony, which is 1 million. Therefore, we have the equation: \(2 * P_0 = P_0 * e^{k*12}\). Dividing by \(P_0\), we get: \(2 = e^{k*12}\). Taking the natural logarithm of both sides, we find: \(ln(2) = 12*k\). Solving for k, we get \(k = ln(2) / 12\). Use a calculator to compute the numeric value of k.

Step 2: Determining when population will reach 4 million

We must solve the equation \(P(t) = 4 * P_0\), or equivalently, \(4 = e^{k*t}\) for \(t\). Doing so, we get \(t = ln(4) / k\). Substituting the previously calculated value of k, compute the value of t. This is when the population will reach 4 million.

Step 3: Determining when population will reach 8 million

Similarly, to determine when the population will reach 8 million, we need to solve the equation \(P(t) = 8 * P_0\), or equivalently, \(8 = e^{k*t}\) for \(t\). Doing so, we get \(t = ln(8) / k\). Again, plugging in our previously found value for k, solve for t. This is when the population will reach 8 million.

Key Concepts

Population Growth

Understanding population growth is essential to grasp how living organisms, such as bacteria, increase in number over time. It involves the study of how the size of the population changes and the factors that cause these changes. An important model to describe population growth, particularly for species that reproduce continuously like bacteria, is the exponential growth model.

In an exponential growth model, a population's growth rate becomes ever more rapid in proportion to the growing total number or size. To put it simply, the larger the population gets, the faster it grows. This type of growth is represented mathematically by the equation \(P(t)=P_0e^{kt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(e\) is the base of the natural logarithm, and \(k\) is the growth constant.

For example, in the exercise given, a colony of bacteria has a population that doubles every 12 hours. This doubling indicates that the bacteria population is experiencing exponential growth. To predict future population sizes or times when the population reaches certain sizes, we use the exponential growth model along with specific initial conditions given in the problem.

Natural Logarithm

The natural logarithm, denoted as \(ln\), is a mathematical function that is the inverse of the exponential function with base \(e\). The natural logarithm of a number \(x\) answers the question: 'To what power must \(e\) be raised, to yield the number \(x\)?' This is particularly useful when solving problems involving exponential equations in population growth.

In population models where we have equations like \(2 = e^{k*12}\), taking the natural logarithm of both sides allows us to isolate and solve for the growth constant \(k\). It effectively 'undoes' the exponential function, turning a potentially difficult equation into a simpler linear one which is easier to solve. That’s why in the step by step solution to the exercise, \(ln(2) = 12k\) is used to find the value of \(k\).

Practical Use of Natural Logarithms

When determining when the population will reach 4 million or 8 million, the natural logarithm is again utilized. By taking the logarithm of both sides, we can solve for time \(t\), providing us with a precise moment when the population reaches the desired size.

Exponential Functions

Exponential functions represent the rate at which a phenomenon grows or decays. In the form \(f(x) = a \times e^{kx}\), the variable \(x\) is the exponent, and the constant \(e\) is the mathematical constant approximately equal to 2.71828. An important trait of an exponential function is that the rate of change is proportional to the function's current value.

In the context of our bacteria population exercise, the exponential function models the rapid growth of the population. As this growth continues over time, it doesn't increase by the same amount but instead multiplies by a certain factor, hence the term 'exponential'.

When using exponential functions to predict scenario outcomes, such as when the population will reach a certain number, it's necessary to understand the concept of the growth constant, \(k\). This constant determines how quickly the population grows. The steps in solving the exercise demonstrate applying the characteristics of exponential functions to solve real-world problems, like predicting the increase in a bacterial colony's population over time.