Problem 52 - Precalculus with Calculus Previews

Question

Find an equation of a sphere that satisfies the given conditions. $$ \text { Center }(-3,1,2) ; \text { passing through the origin } $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Sphere Equation

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by: \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]In this problem, the center is \((-3, 1, 2)\).

Step 2: Determine the Radius

Since the sphere passes through the origin \((0, 0, 0)\), the radius \(r\) is the distance from the center \((-3, 1, 2)\) to the origin. We calculate this distance using the formula for the distance between two points: \[r = \sqrt{(-3 - 0)^2 + (1 - 0)^2 + (2 - 0)^2}\]This simplifies to: \[r = \sqrt{(-3)^2 + 1^2 + 2^2} = \sqrt{9 + 1 + 4} = \sqrt{14}\]

Step 3: Substitute into the Sphere Equation

With the center \((-3, 1, 2)\) and radius \(\sqrt{14}\), substitute these values into the sphere equation:\[(x + 3)^2 + (y - 1)^2 + (z - 2)^2 = (\sqrt{14})^2\]This reduces to:\[(x + 3)^2 + (y - 1)^2 + (z - 2)^2 = 14\]

Key Concepts

Understanding the Center of a Sphere

The center of a sphere is a critical component in defining the shape and position of the sphere in a three-dimensional space. It is represented by a point \(h, k, l\) where \(h, k,\) and \(l\) act as the coordinates of this central point. For any sphere, knowing the center point allows you to calculate the sphere's radius.

In our given problem, the center of the sphere is \((-3, 1, 2)\). These coordinates inform us about the exact location where the center is situated in 3D space. Once we have this point, we can begin to describe and define the entire sphere around this central point. Understanding the center is like finding the heart of the sphere.

Calculating the Radius of a Sphere

The radius is the distance from the center of the sphere to any point on its surface. It's a constant that helps define the size of the sphere.

In our example, the sphere passes through the origin \(0, 0, 0\), and hence the radius can be calculated as the distance from \((-3, 1, 2)\) to the origin. Use the distance formula, a handy tool for finding the straight line distance between two points, to obtain the radius.

Start by finding the differences of the respective coordinates, square them, add them up, and finally take the square root: \[r = \sqrt{(-3 - 0)^2 + (1 - 0)^2 + (2 - 0)^2} = \sqrt{14}\]
This gives us a radius \((r)\) of \(\sqrt{14}\).

The Distance Formula Made Simple

The distance formula is crucial in geometry, especially when dealing with spheres. It helps calculate the distance between two points in space—essential for determining aspects like the radius. This formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

In our scenario, we use the formula to find the radius by setting \(x_1, y_1, z_1\) as \(-3, 1, 2\) (the center) and \(x_2, y_2, z_2\) as \(0, 0, 0\) (the origin). This method is straightforward and effective for solving distances in three-dimensional spaces, providing a consistent approach to finding the radius of spheres or distances in other problems.

Substitution into Equations

Once we have determined both the center \((-3, 1, 2)\) and the radius \(\sqrt{14}\), we can substitute these values into the sphere's equation to define it clearly.

The general equation for a sphere is given by: \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]

Substituting \(h = -3, k = 1, l = 2,\) and \(r^2 = 14\), results in the final equation: \( (x + 3)^2 + (y - 1)^2 + (z - 2)^2 = 14 \).

This step involves systematically replacing placeholders in the formula with specific values, leading to a complete mathematical expression that entirely describes the sphere.