Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Write an inequality that satisfies the description.Outside the circle with radius 3 and center \((0,0)\).

Step-by-Step Solution

Answer

The inequality is \\(x^2 + y^2 > 9\\).

Step 1: Understand the equation of a circle

The equation of a circle centered at the origin \(0,0\) with radius \(r\) is given by \(x^2 + y^2 = r^2\). Here, the radius \(r\) is 3, so the equation is \(x^2 + y^2 = 9\).

Step 2: Formulate the basic inequality

To describe points inside the circle, we would write the inequality \(x^2 + y^2 < 9\), and for points on the boundary, \(x^2 + y^2 = 9\). The question asks for points outside the circle, so the inequality should be changed to \(x^2 + y^2 > 9\).

Step 3: Write the final inequality

The points that lie outside the circle satisfy the inequality \(x^2 + y^2 > 9\). This means any point \(x, y\) for which the sum of the squares of the coordinates is greater than 9 lies outside the circle with radius 3 centered at \(0,0\).

Key Concepts

Inequality Formulation

To understand inequality formulation in this context, we first need to recognize the equation of a circle centered at the origin. The standard equation for a circle is given by \(x^2 + y^2 = r^2\). When we talk about inequalities in terms of circles, we expand this concept to describe areas inside or outside the circle.

For example, in our problem, the circle is centered at \((0,0)\) and has a radius of 3. This means our circle's equation is \(x^2 + y^2 = 9\). Now, using inequalities, we can express various regions:

• Inside the circle: \(x^2 + y^2 < 9\)
• On the circle: \(x^2 + y^2 = 9\)
• Outside the circle: \(x^2 + y^2 > 9\)

By formulating the inequality \(x^2 + y^2 > 9\), we specify that we are interested in points outside the circle. This means the combined squares of the x and y coordinates of any such point must be greater than the square of the radius (9 in this case).

Mathematical Problem Solving

Problem-solving in mathematics often involves understanding which area of a given space you are dealing with. For a problem involving a circle, this means interpreting problems in terms of regions defined by the circle's equation and accompanying inequalities.

This process generally involves:

• Identifying what is needed: In this scenario, we need to find the points outside the given circle.
• Comprehending critical elements: Recognizing the radius and center of the circle provides clarity on the equation and inequalities necessary.
• Applying mathematical procedures: Using inequalities to denote the specific regions, such as inside, on the boundary, or outside.

Solving these types of problems improves logical thinking and your ability to translate words into mathematical expressions and solutions. In this case, framing the mathematical expression \(x^2 + y^2 > 9\) was the key to solving the problem of identifying the area outside the circle.

Coordinate Geometry

Coordinate geometry, or the study of geometry using a coordinate system, allows us to describe geometric shapes, such as circles, with algebraic equations. Through the coordinate plane, every point is marked by a pair of numerical coordinates \((x, y)\) which makes this field highly strategic for solving geometric problems.

Circles in coordinate geometry are expressed via the formula \(x^2 + y^2 = r^2\) for a circle centered at the origin with radius \(r\). In our example:

• The circle is centered at the origin \((0,0)\).
• The radius of the circle is given as 3.
• Thus, the circle's equation is \(x^2 + y^2 = 9\).

With coordinate geometry, we can easily translate problems about points in relation to the circle, such as outside it, into inequalities like \(x^2 + y^2 > 9\). This conversion is invaluable as it allows for clear analysis and provides insights into understanding distances and relationships between the origin and other points in the plane.