Problem 45 - Thomas Calculus

Question

Find equations for the spheres whose centers and radii are given in Exercises \(45-48 .\) $$ (1,2,3) \quad \sqrt{14} $$

Step-by-Step Solution

Answer

The sphere equation is \((x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 14.\)

Step 1: Understanding the Problem

We need to find the equation of a sphere given the center \((1, 2, 3)\) and radius \(\sqrt{14}\).

Step 1: Recall the Sphere Equation

The general equation for a sphere with center \((h, k, l)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\).

Step 2: Identify the Parameters

From the problem, the center \((h, k, l)\) is \((1, 2, 3)\) and the radius \(r\) is \(\sqrt{14}\).

Step 3: Substitute Parameters into the Sphere Equation

Substitute \(h = 1\), \(k = 2\), \(l = 3\), and \(r = \sqrt{14}\) into the sphere equation: \((x - 1)^2 + (y - 2)^2 + (z - 3)^2 = (\sqrt{14})^2\).

Step 4: Simplify the Equation

Calculate \((\sqrt{14})^2 = 14\) to simplify the equation to:\((x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 14.\)

Key Concepts

Understanding the Center of a Sphere

In geometry, the center of a sphere is a key element and is crucial for accurately defining and understanding its location in 3D space. Imagine the center as the anchor point around which the entire sphere is perfectly symmetrical. Every point on the sphere is equidistant from the center.

The center is represented as a point in coordinate geometry, typically denoted by coordinates \( h, k, l \) in a three-dimensional Cartesian coordinate system. This point tells us exactly where the center is located.

• For example, in the exercise, the sphere's center is given as \( (1, 2, 3) \). This means the center is 1 unit along the x-axis, 2 units along the y-axis, and 3 units along the z-axis.
• Using this center in calculations provides a fixed point from which we measure the sphere's size and position.

The center of a sphere is fundamental as it helps in identifying the equation of the sphere and locating it precisely within a space.

Without knowing the center's exact coordinates, determining or graphing the sphere would be impossible.

Determining the Radius of a Sphere

The radius of a sphere in geometry is equally important as the center because it determines the size of the sphere. The radius is the distance from the center of the sphere to any point on its surface. Thus, it controls how "big" or "small" the sphere is.

Every point on the sphere's surface is the same distance from the center. This uniform distance is called the radius. If you're given a center of a sphere \( (h, k, l) \) and the radius \( r \), you can describe the sphere's characteristics quite precisely.

• In the given exercise, the radius is \( \sqrt{14} \).
• To find the actual number, calculate it: \( (\sqrt{14})^2 = 14 \). Hence, using \( \sqrt{14} \) provides the essence of the size calculation without complex multiplication.
• A larger radius indicates a larger sphere, while a smaller radius indicates a smaller sphere.

Recognizing the radius is beneficial when you need to compute the amount of space the sphere occupies or when you graph it alongside other 3D objects.

This measurement is also instrumental in several mathematical and real-world applications like designing spherical objects and in physics for understanding celestial bodies.

Deriving the Equation of a Sphere

The equation of a sphere is a mathematical statement that describes all the points on its surface in three-dimensional space. The standard form of the sphere equation allows you to incorporate both the center and the radius, blending these two aspects into a single formula:

\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]

This formula encompasses the sphere's symmetry around the point center \( (h, k, l) \) and outlines its radius \( r \).

• The subtraction from \( x, y, \) and \( z \) corresponds to locating each coordinate relative to the center.
• The right side, \( r^2 \), demonstrates that all points satisfy the radius square distance requirement.
• Using the original exercise: substitute center coordinates \( (h, k, l) = (1, 2, 3) \) and radius \( r = \sqrt{14} \) into the equation.
• This results in: \( (x - 1)^2 + (y - 2)^2 + (z - 3)^2 = 14 \).

Through this derivation, you've constructed a sphere's complete equation that precisely dictates its spatial definition.

This generalized formula is crucial for creating spheres in programs, simulations, and in real-world design processes where exactness is paramount.