Question

Spread of a Rumor The spread of a rumor in a certain school is modeled by the equation \(P(t)=\frac{300}{1+2^{4-t}}\) where \(P(t)\) is the total number of students who have heard the rumor \(t\) days after the rumor first started to spread. (a) Estimate the initial number of students who first heard the rumor. (b) How fast is the rumor spreading after 4 days? (c) When will the rumor spread at its maximum rate? What is that rate?

Short answer

(a) The initial number of students who heard the rumor is 17. (b) The rate of the rumor's spread after four days is approximately \(300 ln 2\) students per day. (c) The rumor spreads at its maximum rate of approximately \(300 ln 2\) students per day on the fourth day.

Step-by-step solution

Initial number of students who heard the rumor

To find the initial number of students who first heard the rumor, substitute \(t = 0\) into the equation. This gives: \(P(0) = \frac{300}{1 + 2^{4-0}} = \frac{300}{17}\). So, the initial number of students who heard the rumor is 17.

Rate of spreading after four days

The rate of spreading can be found by taking the derivative of the function \(P(t)\) with respect to \(t\). The derivative is given by: \(P'(t) = \frac{300(2^{4-t}) ln 2}{(1+2^{4-t})^2}\). Substituting \(t=4\) into this derivative equation yields: \(P'(4) = \frac{300(2^{4-4}) ln 2}{(1+2^{4-4})^2} = 300 ln 2\). Therefore, the rumor is spreading at a rate of \(300 ln 2\) students per day after four days.

Maximum rate of spread

The maximum rate of spread takes place when the derivative \(P'(t)\) is at its maximum value. This usually occurs at the critical points of the equation, which can be obtained by setting \(P'(t)\) equal to zero and solving for \(t\). Solving this equation, we have: \(P'(t) = 0\), \(300(2^{4-t}) ln 2 = 0\), which gives one critical point at \(t = 4\). Substituting these critical points back into the derivative equation, we find that the maximum rate of spread occurs at \(t = 4\) days with a rate of \(300 ln 2\) students per day.

Key concepts

Exponential Growth and Decay

Understanding the spread of a rumor in a school context can be effectively modeled using the mathematical concept of exponential growth and decay. This concept is crucial in many areas of science and finance, representing processes that increase or decrease at a rate proportional to the amount present. In the case of rumor spreading, the model often assumes that the number of people who know the rumor will grow rapidly at first and then slow down as it reaches more of the population.

The given formula, \(P(t) = \frac{300}{1 + 2^{4-t}}\), captures this behavior by showing how the number of students who have heard the rumor evolves over time. At the start, when only a few people have heard it, it can spread quickly, but as more people become aware of the rumor, there are fewer students to tell, so the growth starts to decelerate. Mathematical modeling of such scenarios provides valuable insights into how information propagates and allows us to predict the pattern of such events using calculus.

Rate of Change

The rate of change of a function refers to the speed at which a quantity represented by that function is changing with respect to time or any other variable. In our example of a spreading rumor, finding how fast the rumor is spreading involves computing the derivative of \(P(t)\), which gives us \(P'(t)\) - the rate of change of the number of students who have heard the rumor at any given time t.

Mathematically, rate of change is synonymous with first derivative. It reveals the instantaneous rate at which one quantity changes with respect to another. In real-life scenarios like the spread of a rumor, understanding the rate of change can help in predicting when the information will reach a certain number of people or when it will spread the most rapidly, which is essential for making informed decisions.

Derivative Application

The application of derivatives in calculus offers powerful tools for solving real-world problems that involve rates of change. When we set out to determine how fast a rumor is spreading after 4 days, we apply the derivative to the function that models the rumor's spread. Taking the derivative of \(P(t)\) gives us the formula for the rate of change, \(P'(t) = \frac{300(2^{4-t})\ln(2)}{(1+2^{4-t})^2}\), which measures how many additional students hear the rumor on any given day t.

To find the maximum rate of spread, we look for the critical points of \(P'(t)\). When solving for the time at which the first derivative is at its maximum, we are effectively finding when the spreading rumor is at its peak – an application that echoes beyond the walls of a school and into fields as diverse as epidemiology and marketing. The derivative, therefore, is not just a mathematical concept but a practical tool for analyzing dynamic processes.