Find the distance between the spheres x^2+y^2+z^2=4 and x^2+y^2+z^2=4 x+4 y+4 z-11.

Name: Problem 43, Chapter 12: Calculus
Uploaded: 2019-07-02T19:13:54-08:00
Duration: 6 min 23 s
Channel: Brittany Knowlton
Description: Find the distance between the spheres x^2+y^2+z^2=4 and x^2+y^2+z^2=4 x+4 y+4 z-11.

step 1 So for question 18, we are given the equation of the sphere, which is x squared plus c squared e

Find the distance between the spheres x^2+y^2+z^2=4 and...

Key Concepts

Sphere Equation Standard Form

A sphere is typically defined by the equation (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) is the center and r is the radius. This format directly displays the intrinsic geometric properties of the sphere, making it easier to analyze its position and size in space.

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation into a perfect square plus a constant. In the context of sphere equations, this technique is crucial for rewriting the given equation in standard form to extract the center and radius, which are not immediately apparent from the general quadratic form.

Distance Formula in 3D

The distance formula in three-dimensional space, given by the square root of the sum of the squares of the differences in each coordinate, calculates the distance between any two points in space. It is fundamental when determining the separation between the centers of two spheres, which is a key step in finding the distance between the surfaces of the spheres.

Distance between Spheres

The distance between two spheres is determined by computing the distance between their centers and then subtracting the sum of their radii. This result gives the minimum distance between any two points on the surfaces of the spheres, and it is essential for assessing how close or far the spheres are from each other.

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