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Magazine Chapter 3: Problem 27
Use a graphing calculator to solve the given equations to the nearest 0.001 $$\sqrt{5 R+2}=3$$
Short Answer
Expert verified
The solution for \( R \) is 1.400.
Step by step solution
01
Understand the Problem
We are given the equation \( \sqrt{5R + 2} = 3 \). The task is to find the value of \( R \) which satisfies this equation using a graphing calculator.
02
Isolate the Radical
The equation \( \sqrt{5R + 2} = 3 \) can be simplified by removing the square root. To do this, square both sides of the equation, resulting in \( 5R + 2 = 9 \).
03
Solve for R
Now, solve for \( R \) by isolating it on one side of the equation. Start with \( 5R + 2 = 9 \). Subtract 2 from both sides to obtain \( 5R = 7 \). Then, divide each side by 5 to get \( R = \frac{7}{5} \).
04
Use a Graphing Calculator
Enter the function \( y = \sqrt{5R + 2} \) and \( y = 3 \) into the graphing calculator. Look for the point of intersection between these two functions. The \( x \)-coordinate of the intersection point is the value of \( R \).
05
Confirm the Solution to the Nearest 0.001
The calculated value for \( R \) should be 1.400 when rounded to three decimal places. Verify this by checking if substituting \( R = 1.400 \) back into the original equation results in a true statement.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Equations
Solving equations is a fundamental concept in mathematics. It involves finding the value of an unknown variable that makes a given equation true. Here's a simplified approach to understand the process:
- Identify the Equation: Look at the mathematical statement. In our case, it's \( \sqrt{5R + 2} = 3 \).
- Understand the Structure: Try to rewrite the equation in a simpler or more familiar form, if possible.
- Manipulate to Solve: Use algebraic operations such as addition, subtraction, multiplication, or division to isolate the variable. For example, squaring both sides of an equation can help eliminate a square root.
Radical Equations
Radical equations are equations in which the variable is inside a radical, often a square root. Solving these requires careful steps to ensure accuracy and avoid mistakes.
- Understanding Radicals: Radicals express roots, such as square roots, cube roots, etc. For example, the square root \( \sqrt{x} \) represents a number which, when squared, gives \( x \).
- Removing Radicals: To solve, you usually need to remove the radical by raising both sides of the equation to the power of the radical. For \( \sqrt{5R + 2} = 3 \), squaring both sides removes the square root, leading to \( 5R + 2 = 9 \).
- Finding Solutions: After removing the radical, solve the resulting simpler equation to find the unknown variable.
Graphical Solutions
Graphical solutions involve plotting equations on a graph to visually find the solution. This method is intuitive and provides a clear picture of what's happening mathematically.
- Plotting Functions: Entering the equation into a graphing calculator will create a visual representation. For example, you plot \( y = \sqrt{5R + 2} \) and \( y = 3 \).
- Finding Intersections: Look for intersections where the curves meet. This point is the solution to the equation as this is where both equations are equal.
- Adjusting Precision: Use the graphing calculator's tools to find the x-coordinate of the intersection to the desired precision, such as 0.001.
Mathematics Education
Mathematics education equips students with problem-solving skills which are applicable across diverse fields and day-to-day issues. Concepts like solving equations and utilizing graphing calculators are important components.
- Understanding Concepts: Education emphasizes not only computational skills but also conceptual understanding, allowing students to apply knowledge in different contexts.
- Use of Technology: Tools like graphing calculators are integral in modern education. They offer a visual and intuitive method to explore mathematical ideas and verify solutions.
- Building Confidence: Learning strategies to tackle different types of equations builds confidence in students' mathematical abilities, preparing them for advanced studies and careers.
