This question aims to find the equation of the sphere centered at (-4, 1, 4) in 3D coordinates and also an equation to describe the intersection of this sphere with a plane z=6.
The question is based on the concepts of a solid geometry. Solid geometry is the part of mathematics geometry that deals with solid shapes like spheres, cubes, cylinders, cones, etc. These shapes are all represented in 3D coordinate systems.
Expert Answer
The given information about this question is as follows:
\[ Center\ of\ Sphere\ c = ( -4, 1, 4) \]
\[ Radius\ of\ Sphere\ r = 3 \]
The general equation for any sphere with center $c = (x_0, y_0, z_0)$ and radius r is given as:
\[ ( x\ -\ x_0 )^2 + ( y\ -\ y_0 )^2 + ( z\ -\ z_0 )^2 = r^2 \]
Substituting the values of this sphere in the general equation, we get:
\[ ( x\ -\ (-4))^2 + ( y\ -\ 1 )^2 + (z\ -\ 4 )^2 = 3^2 \]
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( z\ -\ 4)^2 = 9 \]
This equation represents the sphere, which has a radius of 3, and it is centered at c = (-4, 1, 4).
To find the equation of the intersection of the plane of this sphere, we simply need to put the value of z, which is a plane in the equation of the sphere. Substituting the value of z in the above equation, we get:
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( 6\ -\ 4)^2 = 9 \]
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( 2 )^2 = 9 \]
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + 4 = 9 \]
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 9\ -\ 4 \]
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 5 \]
This represents the intersection of the plane with the sphere.
Numerical Result
The equation of the sphere is calculated to be:
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 + ( z\ -\ 4)^2 = 9 \]
The equation representing the intersection of the sphere with the plane z=6 is calculated to be:
\[ ( x + 4)^2 + ( y\ -\ 1 )^2 = 5 \]
Example
Find the equation of the sphere centered at (1, 1, 1) and radius equal to 5.
\[ Center\ of\ Sphere\ c = ( 1, 1, 1) \]
\[ Radius\ of\ Sphere\ r = 5 \]
Using the general equation of the sphere, we can calculate the equation of the sphere with radius 5 centered at (1, 1, 1).
\[ ( x\ -\ x_0 )^2 + ( y\ -\ y_0 )^2 + ( z\ -\ z_0 )^2 = r^2 \]
Substituting the values, we get:
\[ ( x\ -\ 1 )^2 + ( y\ -\ 1 )^2 + ( z\ -\ 1 )^2 = 5^2 \]
\[ ( x\ -\ 1 )^2 + ( y\ -\ 1 )^2 + ( z\ -\ 1 )^2 = 25 \]
This is the equation of the sphere centered at (1, 1, 1) with a radius of 5 units.