Problem 50
Question
For the following exercises, use this scenario: The equation \(N(t)=\frac{500}{1+49 e^{-0.7 t}}\) models the number of people in a town who have heard a rumor after \(t\) days. How many people started the rumor?
Step-by-Step Solution
Verified Answer
The rumor started with 10 people.
1Step 1: Understanding the Problem
We need to determine the initial number of people who have heard the rumor, which is represented by the function \(N(t)\). This happens at time \(t = 0\).
2Step 2: Substitute Values into the Equation
We substitute \(t = 0\) into the equation \(N(t) = \frac{500}{1+49 e^{-0.7t}}\). This simplifies to \(N(0) = \frac{500}{1+49e^{0}}\) since \(e^{0} = 1\).
3Step 3: Simplify and Solve
Simplify the expression to find \(N(0) = \frac{500}{1+49}\) because \(49 \cdot 1 = 49\). This results in \(N(0) = \frac{500}{50}\).
4Step 4: Calculate the Final Result
Perform the division to find \(N(0) = 10\). This means 10 people initially heard the rumor.
Key Concepts
Rumor SpreadInitial ConditionSubstituting ValuesSimplification Steps
Rumor Spread
Understanding how rumors spread can be likened to an exponential growth model. In this context, the number of people hearing a rumor increases over time in a way that can be mathematically modeled. A common scenario to model such a spread is through using functions analogous to those describing population growth. This growth can happen rapidly and, initially, may involve only a small group of individuals.
The spread of a rumor through a town can be seen as each person informing several others. This continues until the majority of the population becomes aware of it. An exponential growth model is particularly beneficial here, as it allows us to predict and analyze how quickly and extensively a rumor can permeate a community.
The spread of a rumor through a town can be seen as each person informing several others. This continues until the majority of the population becomes aware of it. An exponential growth model is particularly beneficial here, as it allows us to predict and analyze how quickly and extensively a rumor can permeate a community.
Initial Condition
An initial condition in the context of exponential growth models refers to the state of the situation at the very start of the period of interest. For the problem of rumor spreading in a town, the initial condition is crucial as it tells us how many individuals were initially aware of the rumor.
In mathematical modeling, finding the initial condition allows us to set the foundation upon which to build further predictions. In this specific exercise, we determine the initial number of people who have heard the rumor by calculating the value of the model at time, \( t = 0 \). This gives us insight into the starting point of the dynamic process being analyzed.
In mathematical modeling, finding the initial condition allows us to set the foundation upon which to build further predictions. In this specific exercise, we determine the initial number of people who have heard the rumor by calculating the value of the model at time, \( t = 0 \). This gives us insight into the starting point of the dynamic process being analyzed.
Substituting Values
Substituting values is a critical step in solving mathematical problems involving models. In this exercise, we substitute \( t = 0 \) into the given equation. This substitution is the step where we prepare the original model to reflect a specific condition—namely, the initial state in this scenario.
By substituting \( t = 0 \) in the expression \( N(t) = \frac{500}{1+49 e^{-0.7t}} \), the complex part of the equation simplifies since \( e^{0} = 1 \). This reduction is a pivotal simplification that allows us to examine the starting amount of rumor spreaders without the exponential term complicating the calculation.
By substituting \( t = 0 \) in the expression \( N(t) = \frac{500}{1+49 e^{-0.7t}} \), the complex part of the equation simplifies since \( e^{0} = 1 \). This reduction is a pivotal simplification that allows us to examine the starting amount of rumor spreaders without the exponential term complicating the calculation.
Simplification Steps
Simplifying mathematical expressions is essential to deriving manageable forms that are easier to work with. After substituting the values, simplifying the resulting expression \( N(0) = \frac{500}{1+49} \) is the next step.
The expression becomes \( N(0) = \frac{500}{50} \) after recognizing that \( 49 \cdot 1 \) simplifies to \( 49 \).
The expression becomes \( N(0) = \frac{500}{50} \) after recognizing that \( 49 \cdot 1 \) simplifies to \( 49 \).
- Performing the division concludes our simplification, leading to the realization that \( N(0) = 10 \).
- This final step not only simplifies the equation further but also provides a concrete answer to the problem.
Other exercises in this chapter
Problem 49
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$f(x)=1.2 e^{2 x}-0.3, \text { for } f(3)$$
View solution Problem 49
For the following exercises, evaluate the common logarithmic expression without using a calculator. $$2 \log \left(100^{-3}\right)$$
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For the following exercises, solve each equation for \(x\). $$ \log _{3}(3 x)-\log _{3}(6)=\log _{3}(77) $$
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For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. \(f(x)=a b^{x}+d\). $$ -30
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