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Chapter 7: Problem 10

Determine the real root of \(x^{3.5}=80\) with Excel, MATLAB or Math莽ad.

Short Answer

Expert verified
The real root of the equation \(x^{3.5} = 80\) can be found using Excel, MATLAB, or Mathcad as follows: Excel: Using the Goal Seek function, the real root is found to be approximately \(x \approx 3.17\). MATLAB: Using the fsolve function, the real root is found to be approximately \(x \approx 3.17\). Mathcad: Using the solve block, the real root is found to be approximately \(x \approx 3.17\).

Step by step solution

01

Excel Solution

1. Open Excel. 2. In a cell, type the equation as given, but rearrange it to \(f(x) = x^{3.5} - 80\). 3. Choose a starting value for x, let's say x0 = 1. 4. In the cell below x0, apply the Goal Seek function as follows: a. Click on the 'Data' tab. b. Click on the 'What-If Analysis' button. c. Select 'Goal Seek' from the drop-down menu. d. In 'Set Goal', choose the cell containing f(x). e. In 'To value', enter '0'. f. In 'By changing cell', choose the cell containing x0. g. Click on 'OK'. 5. Excel will find the real root and put the value into the cell, x0.
02

MATLAB Solution

1. Open MATLAB. 2. Define the function f(x) = x^3.5 - 80, using the syntax: f = @(x) x.^3.5 - 80; 3. Choose a starting value for x, let's say x0 = 1. 4. To calculate the real root using the fsolve function, type in the following command: real_root = fsolve(f, x0); 5. MATLAB will display the real root in the Command Window.
03

Mathcad Solution

1. Open Mathcad. 2. Define the function f(x) = x^3.5 - 80, using the syntax: f(x) = x^3.5 - 80 3. Choose a starting value for x, let's say x0 = 1. 4. To calculate the real root, use the solve block: a. Click on the 'Functions' tab. b. Click on the 'Define Solve Block' button. c. In the solve block, type the equation: f(x)=0. d. Insert the 'Given' keyword followed by x=x0. e. Finally, insert the 'Find' keyword followed by x. 5. Mathcad will find the real root and display it in the solve block.

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Key Concepts

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

Excel Calculations
Excel is a versatile tool that can be used for more than just spreadsheets and data tracking. It can also be leveraged for numerical methods like finding the roots of equations, such as solving \(x^{3.5} = 80\).
To help Excel find the real root, we utilize its built-in `Goal Seek` feature. This feature allows you to set a formula to zero by adjusting a certain variable incrementally until the desired condition is met.
  • First, you enter the equation as \(f(x) = x^{3.5} - 80\), which reformulates the problem into a typical root-finding scenario.
  • By setting an initial guess, say \(x_0 = 1\), you can then initiate the Goal Seek process.
  • 'Goal Seek' will fine-tune the value of \(x\) by following these adjustments to bring \(f(x)\) closer to zero.
This process turns Excel into a rudimentary yet effective numerical solver for simple root-finding tasks.
MATLAB Programming
MATLAB is a powerful computing environment tailored for mathematical computation and has a built-in approach for solving equations. It's equipped with tools such as `fsolve`, an effective root-finding function.
  • To solve \(x^{3.5} = 80\), we first define the function \(f(x) = x^{3.5} - 80\) within MATLAB via the command: `f = @(x) x.^3.5 - 80`.
  • The `fsolve` function is then used to find the root, initiated with a starting point, e.g., \(x_0 = 1\).
  • `fsolve` iterates to find the value of \(x\) that makes \(f(x) = 0\). This method is particularly robust for complex functions.
MATLAB's advanced capabilities make it ideal for tackling a wide range of numerical problems efficiently.
Mathcad Software
Mathcad provides an intuitive platform for performing mathematical operations that focus on visual clarity and easy manipulation of equations.
  • The process of finding the real root \(x^{3.5} = 80\) involves defining the function as \(f(x) = x^{3.5} - 80\).
  • Mathcad uses a visual solve block, allowing you to input the equation and a starting guess (\(x_0 = 1\)), seamlessly guiding you through solving for the root.
  • With built-in symbolic and numerical solvers, Mathcad efficiently provides results directly in the document.
This approach makes Mathcad a user-friendly and powerful tool for mathematical problem-solving, especially suitable for engineers and scientists who prefer a more interactive layout.
Numerical Methods
Numerical methods are mathematical procedures used to find approximate solutions to equations when analytical solutions are not possible.
They are particularly essential in solving nonlinear equations, like finding the real root of \(x^{3.5} = 80\).
  • The principle behind these methods is to use iterative algorithms to converge close to a solution, which is critical when dealing with complex or transcendental functions that lack straightforward solutions.
  • Diverse tools, including Excel, MATLAB, and Mathcad, implement these algorithms differently, yet share the common goal of approximation.
  • Methods such as `Goal Seek` in Excel or `fsolve` in MATLAB offer practical ways to apply these mathematical concepts in real-world scenarios.
Understanding numerical methods enhances one鈥檚 ability to approach and solve complex mathematical problems efficiently, making them a cornerstone in fields that require extensive computation.

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Most popular questions from this chapter

Use M眉ller's method or MATLAB to determine the real and complex roots of (a) \(f(x)=x^{3}-x^{2}+3 x-2\) (b) \(f(x)=2 x^{4}+6 x^{2}+10\) (c) \(f(x)=x^{4}-2 x^{3}+6 x^{2}-8 x+8\)

Perform the identical MATLAB operations as those in Example 7.7 or use a software package of your choice to find all the roots of the polynomial $$f(x)=(x-6)(x+2)(x-1)(x+4)(x-8)$$ Note that the poly function can be used to convert the roots to a polynomial.

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Determine the roots of the simultaneous nonlinear equations $$\begin{aligned} &y=-x^{2}+x+0.75\\\ &y+5 x y=x^{2} \end{aligned}$$ Employ initial guesses of \(x=y=1.2\) and use the Solver tool from Excel or a software package of your choice.

The velocity of a falling parachutist is given by $$v=\frac{g m}{c}\left(1-e^{-(c / m) t}\right)$$ where \(g=9.8 \mathrm{m} / \mathrm{s}^{2} .\) For a parachutist with a drag coefficient \(c=\) \(14 \mathrm{kg} / \mathrm{s},\) compute the mass \(m\) so that the velocity is \(v=35 \mathrm{m} / \mathrm{s}\) at \(t=8 \mathrm{s}\). Use Excel, MATLAB or Mathcad to determine \(m\)
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