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Chapter 2: Problem 2

Use the rule method to describe the sample space \(S\) consisting of all points in the first quadrant inside a circle of radius 3 with center at the origin.

Short Answer

Expert verified
The sample space \(S\) is the set of all points \((x, y)\) in the first quadrant inside a circle of radius 3 with center at the origin, which can be described by the set \(S = {(x, y)| x \geq 0, y \geq 0, x^2 + y^2 \leq 9}\)

Step by step solution

01

Understanding the Coordinate System

It is important to understand that a 2D coordinate system is divided into four quadrants. The first quadrant is defined as the area where both x and y coordinates are greater than or equal to 0.
02

Understanding circle and its radius

Recall that a circle with the center at the origin and radius 'r' can be described by the equation \(x^2 + y^2 = r^2\). Here, the radius r given is 3.
03

Defining the Sample Space

For a point (x, y) to be inside a circle of radius 3 in the first quadrant, both x and y must be greater than or equal to zero (since it's in the first quadrant) and the sum of the squares of x and y must be less than or equal to 9 (as per the equation of the circle \(x^2 + y^2 = r^2\), with \(r = 3\)). Hence, the sample space is defined as \(S = {(x, y)| x \geq 0, y \geq 0, x^2 + y^2 \leq 9}\).

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Key Concepts

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

Sample Space
In probability and statistics, a *sample space* is the set of all possible outcomes of a particular experiment or situation. When dealing with geometric problems, such as points within certain shapes, the sample space includes all points that satisfy given conditions.
For this exercise, we focus on understanding the sample space within the first quadrant of a plane and inside a circle. To define this, we need to understand both the geometric layout of the circle and how it fits within the coordinate system.
  • The circle has a set radius, and the sample space includes all points that lie within this circle and also satisfy the positional criteria (i.e., are located within the first quadrant).
  • It's essential that all conditions鈥攂eing within the circle and in the correct quadrant鈥攁re met for a point to be considered part of this specific sample space.
  • In our case, every point (x, y) must have both non-negative coordinates and satisfy the circle equation, ensuring it doesn't exceed the circle's boundary defined by the radius.
This precise definition ensures clarity in what outcomes we are considering, which is crucial for solving related problems in probability.
Coordinate System
A coordinate system is a method to define the position of points in a plane using numerical coordinates. The most common type is the 2D Cartesian coordinate system, which divides the plane into four quadrants using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
In this system:
  • The first quadrant is where both x and y coordinates are positive or zero, creating a space that is essentially the upper right section of the plane.
  • In contrast, negative coordinates locate points in the other quadrants, but they are not relevant in this exercise where we focus on the first quadrant.
  • Understanding this division helps visualize how the circle鈥攚ith the center at the origin鈥攆its within these quadrants, specifically identifying which points belong to the first quadrant in relation to our circle.
This foundational knowledge allows us to correctly apply equations like the circle equation only to relevant points, leading to a precise sample space definition.
Circle Equation
The equation of a circle in a coordinate system with the center at the origin (0, 0) and radius 'r' is given by the formula: \(x^2 + y^2 = r^2\).This describes all points (x, y) that form the perimeter of the circle.
To use this formula effectively:
  • Understand that any point satisfyng this equation lies exactly on the circle.
  • Points inside the circle will satisfy \(x^2 + y^2 < r^2\), meaning the sum of the squares of the coordinates is less than the square of the radius.
  • For our specific problem with a circle of radius 3, this becomes \(x^2 + y^2 \leq 9\). This inequality helps define the limits for the sample space inside the circle.
Applying this equation not only helps verify whether a point is inside or on the circle but also helps in defining boundaries for various problems in geometry and probability, as examined in our exercise.

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