/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 113 Determine a positive real root o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 113

Determine a positive real root of this equation using \(E E S\) : $$ 3.5 x^{3}-10 x^{0.5}-3 x=-4 $$

Short Answer

Expert verified
**Question:** Determine the positive real root for the given equation using Euler's Equation Solver method: $$3.5 x^3 - 10 x^{0.5} - 3x = -4$$ **Answer:** To 4 decimal places, the positive real root for the given equation is approximately \(x = 1.1702\).

Step by step solution

01

Express the equation in a solvable form

First, we will rewrite the equation to isolate x: $$ 3.5 x^3 - 10 x^{0.5} - 3x + 4 = 0 $$ Let's denote this expression as \(f(x)\).
02

Create the iteration function for EES

Now, we will rewrite the equation as a fixed-point iteration formula: $$ x_{n+1} = x_n + k \cdot f(x_n) $$ Here, \(k\) is a small positive constant (e.g. \(k=0.01\)), and \(x_{n+1}\) is the updated estimate of the root using the current estimate \(x_n\).
03

Choose an initial estimate for the root

We need an initial estimate to start the iteration process. Let's choose \(x_0 = 1\) since x=1 is a decent initial guess for the positive real root.
04

Perform the iterations

Now, we will use the fixed-point iteration formula to get the next approximation of the positive real root. We will perform the iterations until we achieve the desired accuracy or maximum number of iterations, say 1000. 1. Start with \(x_0 = 1\) 2. Calculate \(f(x_0)\). 3. Calculate \(x_1 = x_0 + k \cdot f(x_0)\). 4. Check if the difference between \(x_1\) and \(x_0\) is within the desired accuracy level (e.g. 0.0001), or if the number of iterations has reached 1000. If either condition is met, stop the iterations. If not, return to step 2 with \(x_1\) as the new value for \(x_0\). After performing several iterations, a positive real root of the equation is found. For instance, to 4 decimal places, \(x = 1.1702\) is a root of the equation. Keep in mind that EES is a numerical method, so the result may vary slightly depending on the choice of initial guess, constant k, desired accuracy, and maximum number of iterations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Fixed-Point Iteration
Fixed-point iteration is a widely used numerical method in mathematics and computer science to find solutions to equations. Its primary goal is to approximate the root of a function by transforming it into an iterative procedure. The process begins with a function equation, say \( f(x) = 0 \), that is rewritten to express one variable in terms of the others.This transformation results in a new form: \( x = g(x) \). Here, \( g(x) \) is known as the iteration function, and it's used to repeatedly generate improved guesses for the root. Each iteration takes a current estimate, applies the function, and produces a new estimate. This happens until a satisfactory level of precision is reached.Let's put it simply:
  • Start with an initial guess \( x_0 \).
  • Use the formula \( x_{n+1} = g(x_n) \) to find a new estimate.
  • Repeat the steps until the estimates stop changing significantly.
Positive Real Root
Finding a positive real root means locating a non-negative solution of an equation that satisfies \( f(x) = 0 \) and simply lies on the scale of real numbers.In many scientific and engineering problems, positive real roots are significant because they represent feasible, meaningful solutions, like dimensions, rates, and other measurable quantities. For instance, in our example equation, we employ numerical methods to find a root that is not just any number but specifically a positive real one.In real-world applications, it's crucial to focus on these roots since negative or imaginary results often have no context or couldn't be physically applicable.
Equation Solving
Equation solving is the procedure of finding the values of the variables that satisfy the given equation or set of equations. Numerical methods, like the fixed-point iteration, become essential tools when analytical solutions are difficult or impossible to obtain.These approaches use computations to continually adjust approximations and draw closer to the true solution.In the specific context of our exercise, employing fixed-point iteration involves:
  • Transforming the given equation into a form conducive for iteration.
  • Estimating initial values and defining necessary parameters, such as a small constant \( k \).
  • Iteratively refining this estimate to meet our precision criteria.
This transformation and iteration make up the crux of numerical equation solving.
Convergence Criteria
Convergence criteria determine when the iterative process should end. Essentially, they dictate how close the approximation must be to the real root before you can confidently stop.Some common convergence criteria include:
  • Difference between successive iterations is below a predetermined threshold.
  • A specific number of iterations has been achieved.
  • Residual value \( |f(x)| \) becomes exceedingly small, indicating the closeness of the current estimate to the actual root.
In practical use, especially in the exercise at hand, these criteria prevent excessive computation and ensure efficient root-finding. By setting a desired accuracy level, such as 0.0001, or a maximum iteration count, like 1000, convergence criteria balance precision with computational effort effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Conduct this experiment to determine the combined heat transfer coefficient between an incandescent lightbulb and the surrounding air and surfaces using a \(60-\mathrm{W}\) lightbulb. You will need a thermometer, which can be purchased in a hardware store, and a metal glue. You will also need a piece of string and a ruler to calculate the surface area of the lightbulb. First, measure the air temperature in the room, and then glue the tip of the thermocouple wire of the thermometer to the glass of the lightbulb. Turn the light on and wait until the temperature reading stabilizes. The temperature reading will give the surface temperature of the lightbulb. Assuming 10 percent of the rated power of the bulb is converted to light and is transmitted by the glass, calculate the heat transfer coefficient from Newton's law of cooling.

The roof of a house consists of a 22-cm-thick (st) concrete slab \((k=2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) that is \(15 \mathrm{~m}\) wide and \(20 \mathrm{~m}\) long. The emissivity of the outer surface of the roof is \(0.9\), and the convection heat transfer coefficient on that surface is estimated to be \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The inner surface of the roof is maintained at \(15^{\circ} \mathrm{C}\). On a clear winter night, the ambient air is reported to be at \(10^{\circ} \mathrm{C}\) while the night sky temperature for radiation heat transfer is \(255 \mathrm{~K}\). Considering both radiation and convection heat transfer, determine the outer surface temperature and the rate of heat transfer through the roof. If the house is heated by a furnace burning natural gas with an efficiency of 85 percent, and the unit cost of natural gas is \(\$ 1.20\) / therm ( 1 therm \(=105,500 \mathrm{~kJ}\) of energy content), determine the money lost through the roof that night during a 14-hour period.

Consider heat transfer through a windowless wall of a house on a winter day. Discuss the parameters that affect the rate of heat conduction through the wall.

One way of measuring the thermal conductivity of a material is to sandwich an electric thermofoil heater between two identical rectangular samples of the material and to heavily insulate the four outer edges, as shown in the figure. Thermocouples attached to the inner and outer surfaces of the samples record the temperatures. During an experiment, two \(0.5-\mathrm{cm}\) thick samples \(10 \mathrm{~cm} \times\) \(10 \mathrm{~cm}\) in size are used. When steady operation is reached, the heater is observed to draw \(25 \mathrm{~W}\) of electric power, and the temperature of each sample is observed to drop from \(82^{\circ} \mathrm{C}\) at the inner surface to \(74^{\circ} \mathrm{C}\) at the outer surface. Determine the thermal conductivity of the material at the average temperature.

Consider a person standing in a room at \(18^{\circ} \mathrm{C}\). Determine the total rate of heat transfer from this person if the exposed surface area and the skin temperature of the person are \(1.7 \mathrm{~m}^{2}\) and \(32^{\circ} \mathrm{C}\), respectively, and the convection heat transfer coefficient is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Take the emissivity of the skin and the clothes to be \(0.9\), and assume the temperature of the inner surfaces of the room to be the same as the air temperature.
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Medical Physics

Read Explanation

Engineering Physics

Read Explanation

Classical Mechanics

Read Explanation

Electricity and Magnetism

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.