Problem 15 - Differential Equations and Linear Algebra

Question

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. At the conclusion of the Super Bowl, the number of fans remaining in the stadium decreases at a rate proportional to the number of fans in the stadium. Assume that there are 100,000 fans in the stadium at the end of the Super Bowl and ten minutes later there are 80,000 fans in the stadium. (a) Thirty minutes after the Super Bowl will there be more or less than 40,000 fans? How do you know this without doing any calculations? (b) What is the half-life (see the previous problem) for the fan population in the stadium? (c) When will there be only 15,000 fans left in the stadium? (d) Explain why the exponential decay model for the population of fans in the stadium is not realistic from a qualitative perspective.

Step-by-Step Solution

Answer

(a) There will be less than 40,000 fans in the stadium 30 minutes after the Super Bowl because the population of fans is decreasing exponentially. (b) The half-life for the fan population is approximately 15.27 minutes. (c) There will be only 15,000 fans left in the stadium approximately 42.98 minutes after the Super Bowl. (d) The exponential decay model is not realistic from a qualitative perspective as it assumes fans will continuously leave the stadium at a rate proportional to the current number of fans, whereas in reality, the rate of fans leaving could change drastically based on certain events or circumstances.

Step 1: Identify the Differential Equation governing the Phenomenon

The population decay model of the fans in the stadium, \(P(t)\), is given by the differential equation: \(\frac{dP}{dt} = kP\) where k is a negative constant.

Step 2: Solve the Differential Equation

To solve the differential equation, we will use separation of variables method. We get: \(\frac{dP}{P} = k dt\) Now, we integrate both sides: \(\int\frac{dP}{P} = \int k dt\) Applying the properties of an indefinite integral, we can rewrite this as: \(\ln|P| = kt + C\) Let's solve for P(t): \(P(t) = e^{kt+C}\) Since \(e^C\) is a constant, we can denote it by A: \(P(t) = Ae^{kt}\)

Step 3: Find the Constant Values Using the Given Information

We are given that at t=0, P(0)=100,000, and at t=10, P(10)=80,000. Let's use these values to find A and k. For t=0 and P(0)=100,000: \(100,000 = Ae^{k*0}\) \(A = 100,000\) For t=10 and P(10)=80,000: \(80,000 = 100,000e^{10k}\) Let's solve for k: \(k = \frac{\ln \frac{80,000}{100,000}}{10} = -\frac{\ln 0.8}{10}\) Now, we have the specific equation for the fan population: \(P(t) = 100,000 \cdot e^{-\frac{\ln 0.8}{10}t}\) (a) Since the function is an exponential decay function, the population will be less than half of the initial population in less than twice the time it took for the initial decrease (10 minutes to 80,000). Thus, 30 minutes after the Super Bowl, there will be less than 40,000 fans. (b) The half-life is defined as the time it takes for the population to be reduced by half. Let's solve for t when P(t)=50,000: \(50,000 = 100,000 \cdot e^{-\frac{\ln 0.8}{10}t}\) Solving for t, we get: \(t = \frac{10\ln 2}{\ln 0.8} \approx 15.27\) minutes (c) To find out when there will be only 15,000 fans left in the stadium, let's solve for t when P(t)=15,000: \(15,000 = 100,000 \cdot e^{-\frac{\ln 0.8}{10}t}\) Solving for t, we get: \(t \approx 42.98\) minutes (d) The exponential decay model for the population of fans in the stadium is not realistic from a qualitative perspective because it assumes that fans will continuously leave the stadium at a rate proportional to the current number of fans. However, in reality, fans may leave in groups or when certain events occur, such as the end of a ceremony, and the rate at which fans leave the stadium could change drastically at any time.

Key Concepts

Differential Equations

In the realm of mathematics, differential equations are powerful tools that describe how various quantities change over time. Think of them as the mathematical equivalent of a time-lapse video, showcasing the dynamic behavior of systems like populations, chemical reactions, or even your savings account. The equation provided in our example, \( \frac{dP}{dt} = kP \), is a classic form of a first-order differential equation. It tells us that the rate of change of a population (or number of fans in a stadium) at any given moment is proportional to its size at that moment. This form of differential equation is prevalent in modeling scenarios where growth or decay is continuously compounded, such as radioactive decay, population studies, or indeed, the rate at which fans leave a stadium post an event. The constant \( k \) in the equation can be positive or negative, indicating growth or decay, respectively. In the case of our Super Bowl example, it's a negative constant, hinting that we're indeed dealing with a diminishing number of fans.

Separation of Variables

The technique of separation of variables is a widely used method to solve first-order differential equations. To apply this method, we rearrange the equation in such a way so that all terms involving the dependent variable (in this case, \( P(t) \) referring to the population) are on one side of the equation, and all terms involving the independent variable (time \( t \) in this context) are on the other side. By integrating both sides, we are essentially summing up these small changes over time to find the general solution to the problem. This technique simplifies the complex dynamics of change into a manageable equation that we can work with, as seen in the Super Bowl example. It allows us to find a specific formula for \( P(t) \) and thus predict crowd numbers at any given time after the game ends, given the appropriate initial conditions.

Half-Life

The term half-life commonly appears in both nuclear physics and any scenario involving exponential decay. Simply put, the half-life of a substance, or in our example, the number of fans in a stadium, is the time required for the amount to reduce to half of its initial value. This concept is especially informative because it provides a constant time frame to compare how quickly various processes decay. In the context of our Super Bowl fans, knowing the half-life helps us understand how fast the stadium is emptying without doing the math each time. It's a crucial tool for making quick estimates or predictions when dealing with any exponentially decaying quantity. In practice, half-life can vary in response to different initial conditions or external influences, although the model assumes ideal conditions where the decay rate remains constant over time.