Problem 47 - Precalculus with Calculus Previews

Question

Find an equation of a sphere that satisfies the given conditions. $$ \text { Center }(-1,4,6) ; \text { radius } \sqrt{3} $$

Step-by-Step Solution

Answer

The equation is \((x + 1)^2 + (y - 4)^2 + (z - 6)^2 = 3\).

Step 1: Understand the General Equation of a Sphere

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

Step 2: Substitute the Known Values

In this exercise, the center \((h, k, l)\) is given as \((-1, 4, 6)\) and the radius \(r\) is given as \(\sqrt{3}\). Substitute these values into the general equation:\[ (x + 1)^2 + (y - 4)^2 + (z - 6)^2 = (\sqrt{3})^2 \].

Step 3: Simplify the Equation

Now, simplify the right-hand side of the equation. Since \((\sqrt{3})^2 = 3\), the equation of the sphere becomes: \[ (x + 1)^2 + (y - 4)^2 + (z - 6)^2 = 3 \].

Key Concepts

Sphere

A sphere is a perfect three-dimensional shape. Imagine the outer surface of a ball—this is what a sphere looks like. Every point on the surface of a sphere is at an equal distance from a fixed point inside, called the center.

This is an important characteristic that sets spheres apart from other shapes. Unlike cubes or pyramids, a sphere doesn’t have edges or corners. It’s completely smooth and round, making it unique in the world of geometry.

• Every point on the surface is equally distant from the center.
• No edges or corners exist on a sphere.
• This shape is symmetrical from every direction.

The equation of a sphere helps describe its size and position in a three-dimensional space, giving us an analytical way to understand and work with this shape.

Radius

The radius of a sphere is the distance from its center to any point on its surface. Think of it as a spoke on a wheel—it's like the line stretching from the center to the wheel's edge.

The radius is crucial because it defines the size of the sphere. The larger the radius, the bigger the sphere. It's used in the formula of the sphere's equation, which allows you to calculate various aspects, such as its surface area or volume when needed.

• Always a straight line from the center to the outer surface.
• Half the length of the diameter (distance across the sphere through the center).
• Essential for the sphere's equation: \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\).

The radius in the provided exercise is given as \(\sqrt{3}\), showing how it can sometimes be represented in a square root form to maintain precision.

Center of a Sphere

The center of a sphere is the fixed point inside the sphere from which every point on the surface is equidistant. It's like the hub of a wheel where all the spokes radiate outward.

Knowing the center's coordinates is vital because it allows you to use the sphere's equation to find out more about its position and dimensions.

• The coordinates \((h, k, l)\) define the center in a mathematical expression.
• In the equation of a sphere, \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), \(h, k, l\) reflect the center’s position.
• The exercise states the center as \((-1, 4, 6)\), indicating its specific location in a coordinate system.

The center is more than just a point; it's a critical element that affects how the sphere is situated in space, guiding calculations and representations of this fascinating geometric shape.