Problem 32 - Calculus for Biology and Medicine

Question

Draw a closed disk with radius 3 centered at \((2,0)\) in the \(x-y\) plane, and give a mathematical description of this set.

Step-by-Step Solution

Answer

The closed disk is described by \((x - 2)^2 + y^2 \leq 9\).

Step 1: Understanding the Problem

The task is to draw a closed disk on the coordinate plane with the given specifications: a center at the point (2, 0) and a radius of 3 units. A closed disk includes all points that are within or on the circle defined by these parameters.

Step 2: Understanding a Closed Disk Equation

The mathematical equation for a circle (or disk boundary) with center (a, b) and radius (r) is \((x - a)^2 + (y - b)^2 = r^2\). For a closed disk, the inequality is \((x - a)^2 + (y - b)^2 \leq r^2\), which covers all interior points including the boundary.

Step 3: Substituting Given Values

In this case, the center is (2, 0) and the radius is 3. Substitute these values into the closed disk inequality: \((x - 2)^2 + (y - 0)^2 \leq 3^2\). This simplifies to \((x - 2)^2 + y^2 \leq 9\).

Step 4: Graphical Representation

Draw the circle with center at (2, 0) and a radius of 3. This involves plotting points at a distance of 3 from the center in all directions, forming a circular or disk shape. Shade the entire area including the boundary to represent all points described by the inequality.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point in this plane can be represented as a pair of numbers \( (x, y) \), where \( x \) corresponds to a location along the x-axis and \( y \) corresponds to a location along the y-axis. This system allows us to precisely locate points and shapes within the plane by specifying their coordinates.

• The origin, denoted by the point \( (0, 0) \), is where the x-axis and y-axis intersect.
• Quadrants divide the plane into four sections: I, II, III, and IV. Each quadrant represents a unique set of positive and negative values for \( x \) and \( y \).
• Coordinates are written in the format \( (x, y) \), assessing position relative to the origin.

Understanding how to navigate and plot on the coordinate plane is essential for graphing equations and visualizing shapes like disks.

Mathematical Inequality

In mathematics, an inequality is a relation that holds between two values when they are different. The symbols \( <, >, \leq \, \geq \, eq \) are used to represent inequalities and indicate how one value compares to another:

• \( < \, > \) signify that numbers are less than or greater than others.
• \( \leq \, \geq \) denote numbers less than or equal to, or greater than or equal to others.

For a closed disk in the coordinate plane, the specific inequality used is \( (x - a)^2 + (y - b)^2 \leq r^2 \). This inequality encompasses all the points within and on the boundary of a circle centered at \( (a, b) \) with a radius \( r \). The inequality \( \leq \) ensures points on the boundary are included, forming a complete disk. Recognizing and applying these inequalities allows us to mathematically describe shapes and regions.

Graphical Representation

Graphical representation involves translating mathematical equations or inequalities into shapes and images on the coordinate plane. When representing a closed disk graphically, we interpret the mathematical description into a visual format, helping to better understand spatial relationships and set inclusions.

• Plot the center point \( (2, 0) \) as provided in the exercise.
• Use the radius, \( r = 3 \), to measure all directions from the center, forming the circle or boundary of the disk.
• Shade the area inside the circle, including the boundary itself, to indicate the set of points fulfilling the inequality \( (x - 2)^2 + y^2 \leq 9 \).

This visual clarity supports problem-solving and helps in verifying mathematical solutions and interpretations of geometric shapes.