Problem 11 - Precalculus Mathematics for Calculus

Question

Find an equation of a sphere with the given radius \(r\) and center \(C\). $$r=5 ; \quad C(2,-5,3)$$

Step-by-Step Solution

Answer

The sphere's equation is \((x - 2)^2 + (y + 5)^2 + (z - 3)^2 = 25\).

Step 1: Understand the Sphere's Equation

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). This equation stems from the distance formula in three dimensions.

Step 2: Identify the Values

We are given the center \(C(2, -5, 3)\) and the radius \(r = 5\). Thus, we identify \(h = 2\), \(k = -5\), and \(l = 3\). For the radius, we have \(r = 5\).

Step 3: Substitute Values into the Sphere's Equation

Now substitute the values of \(h\), \(k\), \(l\), and \(r\) into the sphere's equation: \((x - 2)^2 + (y + 5)^2 + (z - 3)^2 = 5^2\).

Step 4: Simplify the Equation

Compute the square of the radius: \[5^2 = 25\].Thus, the equation becomes: \[(x - 2)^2 + (y + 5)^2 + (z - 3)^2 = 25\].

Key Concepts

Three-Dimensional Geometry

Three-dimensional geometry involves studying shapes and figures in a 3D space. This includes objects that have volume and three dimensions: length, width, and height. Unlike two-dimensional shapes like squares and circles, three-dimensional shapes include spheres, cubes, and pyramids, which occupy space. In three-dimensional geometry, each point in space is defined by three coordinates:

• (x, y, z) - where 'x' represents the position along the horizontal axis, 'y' along the vertical axis, and 'z' the depth or height.
• These coordinates help in locating points, drawing shapes, and understanding spatial relationships.

Understanding 3D geometry is essential because our world is three-dimensional. Recognizing how geometric concepts apply in three dimensions enhances spatial visualization skills. This is particularly crucial when analyzing the properties and positions of three-dimensional objects like spheres.

Distance Formula

The distance formula in three dimensions is an extension of the Pythagorean theorem and is used to determine the distance between two points in space. If you know the coordinates of the two points, say

• Point A \( (x_1, y_1, z_1) \)
• Point B \( (x_2, y_2, z_2) \)

you can calculate the distance between them using:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]This formula allows us to compute the straight-line distance, or the shortest path, between two points in 3D space. In the case of a sphere, the formula translates into describing the distance from any point on the surface to its center. This understanding is foundational to deriving the sphere's equation and is integral to how three-dimensional objects are mapped and analyzed mathematically.

Radius and Center of a Sphere

The sphere is a perfectly symmetrical three-dimensional object, and its description is mathematically elegant. A sphere's size and position are described using two critical components:

• Center: The center of a sphere is a fixed point in space, usually given by the coordinates \((h, k, l) \).
• Radius: The radius \( r \) is the constant distance from the center to any point on the surface of the sphere.

Given these, the equation of the sphere is \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). This equation essentially defines all points \( (x, y, z) \) that are exactly the radius away from the center point \((h, k, l) \). In practice, to define a specific sphere, you substitute the given values of radius and center into this equation. This process yields the formula to describe a sphere precisely, highlighting how the radius and center determine the sphere's position and size in three-dimensional space.