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Chapter 5: Problem 12

The equation \(x^{2}+y^{2}=4\) is a circle centered at _________ with radius ___________ the solution set to the inequality \(x^{2}+y^{2}<4\) represents the set of points (inside/outside) the circle \(x^{2}+y^{2}=4\).

Short Answer

Expert verified
The center is (0, 0) and the radius is 2. The inequality represents points inside the circle.

Step by step solution

01

Identify the Equation of the Circle

Recognize that the equation given, \({x^{2}+y^{2}=4},\)} represents a circle in the form \({(x-h)^{2}+(y-k)^{2}=r^{2}\)}. Compare this with the standard form to identify the center and radius.
02

Determine the Center

From \({x^{2}+y^{2}=4}\), we can see that the equation is in the form \({(x-0)^{2}+(y-0)^{2}=4}.\) Therefore, the center of the circle is at (0, 0).
03

Determine the Radius

Since \({r^{2}=4}.\), then \({r=\text{sqrt}(4)}\) which simplifies to \({r=2}.\) Therefore, the radius is 2.
04

Interpret the Inequality

The inequality \({x^{2}+y^{2}<4}\)} represents all the points inside the circle \({x^{2}+y^{2}=4}\), since it includes all points with a distance less than 2 from the center (0, 0).

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Key Concepts

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

Standard form of circle
To understand circle equations, it helps to start with the standard form of a circle equation. This form highlights the circle's center and its radius.
The standard form of a circle equation is: \[{(x-h)^{2}+(y-k)^{2}=r^{2}}\]
In this form:
  • \
Center and radius identification
Let鈥檚 dive into identifying the center and radius of the circle from the standard form.
The given equation in the example is \[{x^{2}+y^{2}=4}}\].
Using the standard form notation, we can rewrite it as: \
Graphing inequalities
Inequalities involving circles can be tricky, but breaking them down step by step makes them easier to understand.
In this example, the inequality given is \[{x^{2}+y^{2}<4}}\].
This inequality represents all the points inside the circle, because...

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Most popular questions from this chapter

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 8=4 \lambda x \\ 2=2 \lambda y \\ 2 x^{2}+y^{2}=9 \end{array} $$

A large department store needs at least 3600 labor hours covered per week. It employs full-time staff \(40 \mathrm{hr} / \mathrm{wk}\) and part-time staff \(25 \mathrm{hr} / \mathrm{wk}\). The cost to employ a full-time staff member is more because the company pays benefits such as health care and life insurance. $$ \begin{array}{|l|c|c|} \hline & \text { Hours per Week } & \text { Cost per Hour } \\ \hline \text { Full time } & 40 \mathrm{hr} & \$ 20 \\ \hline \text { Part time } & 25 \mathrm{hr} & \$ 12 \\ \hline \end{array} $$ The store manager also knows that to make the store run efficiently, the number of full-time employees must be at least 1.25 times the number of part- time employees. a. Determine the number of full-time employees and the number of part-time employees that should be used to minimize the weekly labor cost. b. What is the minimum weekly cost to staff the store under these constraints?

Write a system of inequalities that represents the points inside the triangle with vertices \((-4,-4),(1,1),\) and (5,-1).

a. Sketch the lines defined by \(y=2 x\) and \(y=-\frac{1}{2} x+5\) b. Find the area of the triangle bounded by the lines in part (a) and the \(x\) -axis.

Solve the system using any method. $$ \begin{array}{l} 5(2 x+y)=y-x-8 \\ x-\frac{3}{2} y=\frac{5}{2} \end{array} $$
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