91Ó°ÊÓ

Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let \(f(t)\) be the fraction of manufacturers that have complied with the regulations within \(t\) months. Note that \(0 \leq f(t) \leq 1 .\) Suppose that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, with constant of proportionality \(k=.1\). (a) Construct a differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=3\) to estimate the fraction of companies that comply with the regulations within the first 3 months. (c) Solve the differential equation in part (a) and compute \(f(3).\) (d) Compare the answers in parts (b) and (c) and approximate the error in using Euler's method.

Step by step solution

01

Construct the Differential Equation

Given that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, the differential equation can be expressed as: \[ \frac{df}{dt} = k(1 - f(t)) \] Given that the constant of proportionality \( k = 0.1 \), we can write the differential equation as: \[ \frac{df}{dt} = 0.1 (1 - f(t)) \]

02

Euler's Method with n=3

Euler's method involves estimating the value of \( f(t) \) over small increments of time. The general formula for Euler's method is: \[ f_{n+1} = f_n + h \cdot f'(t_n) \] Given that \( n = 3 \) and assuming a step size \( h = 1 \) month for simplicity, we start with \( f(0) = 0 \) (assuming no compliance at the start).1. For the first month: \[ f(1) = f(0) + 1 \cdot 0.1(1 - f(0)) = 0 + 0.1 = 0.1 \]2. For the second month: \[ f(2) = f(1) + 1 \cdot 0.1(1 - f(1)) = 0.1 + 0.1(1 - 0.1) = 0.1 + 0.09 = 0.19 \]3. For the third month: \[ f(3) = f(2) + 1 \cdot 0.1(1 - f(2)) = 0.19 + 0.1(1 - 0.19) = 0.19 + 0.081 = 0.271 \] So, the Euler's method estimate after 3 months is approximately \( f(3) \approx 0.271 \).

03

Solve the Differential Equation

To solve the differential equation \( \frac{df}{dt} = 0.1 (1 - f(t)) \), separate the variables and integrate: \[ \int \frac{1}{1 - f(t)} \, df = 0.1 \int dt \]\[ -\ln|1 - f(t)| = 0.1t + C \] Exponentiating both sides, we get: \[ 1 - f(t) = Ce^{-0.1t} \] Setting the initial condition \( f(0) = 0 \), then \( 1 - 0 = Ce^0 \Rightarrow C = 1 \). Therefore, \( 1 - f(t) = e^{-0.1t} \), so \[ f(t) = 1 - e^{-0.1t} \] To find the fraction of companies that comply in 3 months, substitute \( t = 3 \): \[ f(3) = 1 - e^{-0.3} \approx 1 - 0.7408 = 0.2592 \]

04

Compare Euler's Method with Exact Solution

The estimated fraction using Euler's method was \( 0.271 \), and the exact calculation yields \( 0.2592 \). The error can be calculated as: \[ \text{Error} = |0.271 - 0.2592| = 0.0118 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's method

Euler's method is a simple yet powerful tool used to approximate solutions to differential equations. It works by using a step-by-step process to estimate the value of a function based on its derivative. In this method, the formula for the next value, given the current value, is: \( f_{n+1} = f_n + h \times f'(t_n) \), where \( h \) is the step size. For the toy-manufacturing problem, we used a step size of 1 month to estimate the compliance rate over 3 months. Starting with \( f(0) = 0 \) and applying Euler's method, we found \( f(1) = 0.1 \), \( f(2) = 0.19 \), and \( f(3) = 0.271 \). This method is particularly useful when solving complex differential equations that can't be easily handled analytically.

Proportionality constant

A proportionality constant acts as a scaling factor to relate the rate of change of a quantity to its current value or the value of another quantity. In the given problem, the compliance rate of toy manufacturers was stated to be proportional to the fraction of companies that have not yet complied. The constant of proportionality \( k \) was given as 0.1. This means the rate at which companies comply depends significantly on the number of companies yet to comply. Mathematically, this relationship was expressed as \( \frac{df}{dt} = k(1 - f(t)) \). The presence of the proportionality constant ensures that as more companies comply, the rate of new compliance decreases in proportion, capturing a realistic dynamic.

Separable Differential Equations

Separable differential equations are special types where variables can be separated on different sides of the equation, allowing easier integration. For our exercise, the differential equation formed was \( \frac{df}{dt} = 0.1 (1 - f(t)) \). We separated the variables as follows: \( \int \frac{1}{1 - f(t)} \, df = 0.1 \int dt \). This separation allows for direct integration of both sides: \[ -\ln|1 - f(t)| = 0.1t + C \]. After integrating and solving for the integration constant using the initial condition \( f(0) = 0 \), we obtained \( f(t) = 1 - e^{-0.1t} \) as the analytical solution. Separable differential equations simplify the process of finding exact solutions by making integration straightforward.

Error approximation

Error approximation measures how close our estimated solution (via methods like Euler's method) is to the exact solution. In our case, Euler's method provided an estimated compliance fraction of \( f(3) \approx 0.271 \) after 3 months. Solving the differential equation exactly, we found the compliance fraction to be \( f(3) \approx 0.2592 \). The error in Euler's method is calculated as: \[ |0.271 - 0.2592| = 0.0118 \]. This small error indicates that while Euler's method gives a good approximation, it is not exact. Such approximations are useful in practical scenarios where exact solutions are difficult to compute. By understanding the error margin, we can gauge the reliability of our approximations.