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Chapter 9: Problem 48

Many nonlinear systems cannot be solved algebraically, so graphical analysis is the only way to determine the solutions of such systems. Use a graphing calculator to solve each nonlinear system. Give \(x\) - and \(y\) -coordinates to the nearest hundredth. $$\begin{aligned} &y=\sqrt[3]{x-4}\\\ &x^{2}+y^{2}=6 \end{aligned}$$

Short Answer

Expert verified
The solutions are approximately (3.67, -1.45) and (4.33, 1.45).

Step by step solution

01

Understand the Equations

The system consists of two equations. The first is a cube root function: \(y = \sqrt[3]{x-4}\). The second is a circle equation: \(x^{2} + y^{2} = 6\).
02

Graph the Cube Root Function

Use a graphing calculator to plot the cube root function \(y = \sqrt[3]{x-4}.\) This graph will look like an 'S' curve centered around the point (4, 0).
03

Graph the Circle

Next, plot the circle given by the equation \(x^{2} + y^{2} = 6.\) This is a circle centered at the origin (0, 0) with radius \(\sqrt{6} \approx 2.45.\)
04

Find the Intersection Points

Using the graphing calculator, find the points where the cube root function and the circle intersect. These points are where the solutions to the system lie.
05

Determine the Coordinates

Zoom in or use the trace feature of the calculator to determine the coordinates of the intersection points to the nearest hundredth.
06

Report the Solutions

Write down the coordinates of the intersection points. These are the solutions to the system of equations.

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Key Concepts

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

Graphing Calculator
Graphing calculators are powerful tools for solving systems of equations, especially when they are nonlinear. These calculators allow you to visualize functions and their intersections, which might be difficult or impossible to solve algebraically.
To use a graphing calculator:
  • Input each equation into the calculator.
  • Set the appropriate viewing window to capture the critical parts of the graphs.
  • Use features like zoom and trace to precisely identify points of intersection.
By leveraging these tools, you can accurately determine where graphs intersect, which translates to solving the system of equations.
Cube Root Function
The cube root function, represented as \(y = \sqrt[3]{x-a}\), is a type of nonlinear function. The graph of this function forms an 'S' shaped curve. In the given system, the function is \(y = \sqrt[3]{x-4}\), which means it shifts the basic cube root function 4 units to the right.
When plotting this on a graphing calculator, you'll notice its unique shape:
  • It passes through the point \( (4, 0) \).
  • The curve goes from bottom left to top right.
  • It doesn't have any maximum or minimum points, just inflection points.
This function is continuous and infinite in both directions, making it crucial to understand its behavior for accurate graphing and analysis.
Circle Equation
A circle equation is generally represented as \(x^{2} + y^{2} = r^{2}\), where \(r\) is the radius of the circle. For the system in question, the circle equation is given by \(x^{2} + y^{2} = 6\).
Here are key aspects to note:
  • The center of the circle is at the origin \( (0,0) \).
  • The radius \( r = \sqrt{6} \approx 2.45 \).
  • It is a closed curve, symmetric about both axes.
When graphing the circle, ensure the radius and center are clearly captured to properly visualize where it intersects with other functions like the cube root function.
Intersection Points
Intersection points are where the graphs of two equations meet. For solving a system graphically, these points represent the solutions.
When you graph \( y = \sqrt[3]{x-4} \) and \( x^{2} + y^{2} = 6 \):
  • Look for points where the 'S' curve and the circle touch or cross each other.
  • Use the trace or zoom feature on your graphing calculator to pinpoint these coordinates.
  • The solutions are often in pairs like \( (x,y) \).
Accurate determination of these intersections is vital as they give the comprehensive solutions to the nonlinear system.
Solutions to Systems of Equations
The solution to a system of equations is the set of coordinates that satisfy all equations simultaneously. For nonlinear systems:
  • These solutions are the intersection points on the graph.
  • An analytical approach may not always work, making graphical solutions practical.
  • Verify the coordinates by substituting back into the original equations to ensure accuracy.
In the current exercise, after graphing, we zoom and trace the graphs to find the intersection points. The final solutions are those coordinates rounded to the nearest hundredth as they fit both the cube root and circle equations.

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Most popular questions from this chapter

$$ A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad B=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right], \quad \text { and } \quad C=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] $$ where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B\) for any real number \(c\)

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} -2 x-2 y+3 z &=4 \\ 5 x+7 y-z &=2 \\ 2 x+2 y-3 z &=-4 \end{aligned}$$

Solve each problem. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \(\$ 351 .\) How many of each denomination of bill are there?

Find the equation of the circle passing through the given points. $$(2,1),(-1,0), \text { and }(3,3)$$

Solve each system for \(x\) and y using Cramer's rule. Assume a and b are nonzero constants. $$\begin{array}{l} b x+y=a^{2} \\ a x+y=b^{2} \end{array}$$
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