Question

Find the equation of the sphere centered at (-9, -8, 9) with a radius of 4.

a) Provide a step-by-step explanation of the process used to derive the equation of the sphere.
b) Include the general form of the equation for a sphere in three-dimensional space and explain how the given center coordinates and radius are incorporated into this equation.
c) Discuss the significance of each parameter in the resulting equation and how it contributes to the description of the sphere.

Ensure clarity and detail in your response to showcase a comprehensive understanding of the equation of a sphere.

Answer 1

The equation of a sphere is (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) are the coordinates of the center and r is the radius. For this particular sphere, the equation is (x + 9)^2 + (y + 8)^2 + (z - 9)^2 = 16.

Step-by-step explanation:

The equation of a sphere in three-dimensional space is given by:

(x - h)2 + (y - k)2 + (z - l)2 = r2

where (h, k, l) are the coordinates of the center of the sphere and r is the radius.

In this case, the sphere is centered at (-9, -8, 9) and has a radius of 4. So, we can substitute these values into the general equation to get the equation of this particular sphere:

(x + 9)2 + (y + 8)2 + (z - 9)2 = 16

The parameters in the resulting equation have the following significance:

- (x, y, z) represent any point on the surface of the sphere.

- (h, k, l) represent the coordinates of the center of the sphere.

- r represents the radius of the sphere.