Problem 14 - Precalculus Mathematics for Calculus

Question

Find an equation of a sphere with the given radius \(r\) and center \(C\). $$r=\sqrt{11} ; \quad C(-10,0,1)$$

Step-by-Step Solution

Answer

The equation of the sphere is \((x + 10)^2 + y^2 + (z - 1)^2 = 11\).

Step 1: Identify the formula for a sphere

The equation of a sphere with center \(C(x_0, y_0, z_0)\) and radius \(r\) is \((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2\).

Step 2: Substitute the known center coordinates

Given the center \(C(-10, 0, 1)\), substitute these values into the formula: \((-10, 0, 1)\), it becomes \((x + 10)^2 + (y - 0)^2 + (z - 1)^2 = r^2\).

Step 3: Substitute the known radius

The given radius is \(r = \sqrt{11}\), which means \(r^2 = 11\). Substitute this into the equation: \((x + 10)^2 + (y - 0)^2 + (z - 1)^2 = 11\).

Step 4: Write the final equation

Combine all substitutions to write the final equation for the sphere: \( (x + 10)^2 + y^2 + (z - 1)^2 = 11 \).

Key Concepts

Radius of a Sphere

The radius of a sphere is a key element in defining its equation. It is the distance from the center of the sphere to any point on its surface. In terms of mathematics, if you know the radius, you know how large the sphere is.
To pinpoint the radius precisely, we often square it in calculations. For instance, if a sphere's radius is given as \( r = \sqrt{11} \), its square is \( r^2 = 11 \).
This simplified representation is crucial in the sphere equation, as it helps describe the entire surface in terms of distance from the center.

Center of a Sphere

The center of a sphere is another fundamental part of its equation. It acts as the reference point from which all parts of the sphere are equidistant.
In the context of three-dimensional space, the center of a sphere is typically represented by coordinates \( C(x_0, y_0, z_0) \). These coordinates explain where the center lies along the x, y, and z axes.
For example, if a sphere's center is at \( C(-10, 0, 1) \), it means the center point is located 10 units left on the x-axis, perfectly centered at 0 on the y-axis, and 1 unit above the origin on the z-axis.

Coordinate Substitution

Coordinate substitution is essential to formulating the sphere's equation. It involves replacing the variables in the general formula with specific values derived from the sphere's characteristics.
Given a sphere's center \( C(x_0, y_0, z_0) \), we plug these into the formula \((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2\).
For instance, if the center is \( C(-10, 0, 1) \), you substitute to get: \((x + 10)^2 + (y - 0)^2 + (z - 1)^2\). This step personalizes the equation to your specific sphere.

Equation Derivation

Deriving the equation of a sphere involves integrating all known parameters into a structured format that defines every possible point on the sphere's surface.
Initially, you start with the fundamental formula \((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2\). Here, you insert the specific center coordinates for \(x_0, y_0, z_0\), and square the radius.
For the sphere with center \((-10, 0, 1)\) and radius \(\sqrt{11}\), the completed equation reads \((x + 10)^2 + y^2 + (z - 1)^2 = 11\). This captures the sphere's distinctive position and size.