Problem 18 - Calculus Single Variable

Question

A circle is described in words. Give its Cartesian equation. The circle with center (3,0) and diameter 8

Step-by-Step Solution

Answer

The Cartesian equation of the circle is \((x - 3)^2 + y^2 = 16\).

Step 1: Understanding the Circle's Properties

The center of the circle is given as (3, 0). The diameter is provided as 8, which implies the radius of the circle is half of the diameter. Calculate the radius: \[ r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \]

Step 2: Formulating the General Equation of a Circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] In this example, \(h = 3\), \(k = 0\), and \(r = 4\).

Step 3: Plugging Values into the Equation

Substitute the values of \(h\), \(k\), and \(r\) into the general circle equation: \[ (x - 3)^2 + (y - 0)^2 = 4^2 \] Simplifying this equation yields: \[ (x - 3)^2 + y^2 = 16 \]

Step 4: Writing the Final Cartesian Equation

The final form of the Cartesian equation of the circle is thus: \[ (x - 3)^2 + y^2 = 16 \] This represents the circle with the given center and radius derived from the diameter.

Key Concepts

Circle Geometry

Circle geometry is a fascinating area of mathematics that explores the properties and relationships of circles. A circle is a set of all points in a plane that are equidistant from a fixed point called the center. This distance is known as the radius. Circles are fundamental shapes in geometry, revered for their perfect symmetry and used in various fields such as architecture, engineering, and art.

Key characteristics of a circle include its center, radius, diameter, and circumference. Understanding these elements allows us to create equations and solve geometrical problems. The circle's equation, in particular, serves as a mathematical representation that encapsulates all these properties into a formulaic expression.

Radius Calculation

The radius of a circle is the distance from its center to any point on its boundary. In problems where the diameter is given, calculating the radius becomes crucial. The diameter of a circle is twice the radius, represented by the equation:

• \[ ext{Diameter} = 2 imes ext{Radius} \ ext{or equivalently} \ ext{Radius} = \frac{ ext{Diameter}}{2} \]

Knowing how to find the radius quickly is essential as it often serves as a stepping stone to more complex calculations like formulating an equation or computing the area and circumference. In our exercise, with a given diameter of 8, the radius was calculated using the formula above, resulting in a radius of 4.

Diameter and Radius

The diameter and radius are closely linked concepts that describe the size of a circle. The diameter is the longest distance across the circle, passing through the center. It is crucial for describing how large or small a circle is.

On the other hand, the radius is a more fundamental measure, often used to perform calculations in circle geometry, such as finding the area (\( A = \pi r^2 \)) or the circumference (\( C = 2\pi r \)).

In practical applications, knowing both the diameter and radius helps in visualizing the circle's scale. Understanding these terms makes it easier to grasp circle-related equations and apply them to solve problems.

Equation Formulation

Formulating the equation of a circle is a straightforward process once you understand its properties. The standard Cartesian equation of a circle is derived from its center and radius, expressed as:

• \[ (x-h)^2 + (y-k)^2 = r^2 \]

where \((h, k)\) is the center and \(r\) is the radius.

In this format,

• \( (x-h)\) and \((y-k)\) describe the horizontal and vertical distances from any point on the circle to its center.
• The equation itself maintains that these distances always equate to the radius squared, ensuring a perfectly round shape.

For the exercise, substituting the center \((3,0)\) and radius \(4\), we arrived at the simplified equation:

• \[ (x - 3)^2 + y^2 = 16 \]

This final equation beautifully encapsulates the circle's characteristics, providing a comprehensive way to analyze and solve circle-related problems in Cartesian coordinates.