Math Problem Statement
Find the volume of the solid that lies inside both of the spheres, x^2 + y^2 + z^2 +4x - 2y + 4z + 5 = 0 and x^2 + y^2 + z^2 =4
Solution
To find the volume of the solid inside both spheres, we need to analyze the equations of the two spheres and their intersection. The equations of the spheres are:
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First sphere:
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Second sphere:
Step 1: Completing the square for the first sphere's equation
To simplify the first equation into the standard form of a sphere, we complete the square for the , , and terms:
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For , complete the square:
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For , complete the square:
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For , complete the square:
Substitute these into the original equation of the first sphere:
Simplify:
Thus, the first sphere has center and radius .
Step 2: Understanding the geometry of the intersection
- The second sphere is centered at the origin and has radius .
- The first sphere is centered at and also has radius .
Since the two spheres have the same radius and their centers are units apart, the spheres intersect.
Step 3: Volume of the intersection of two spheres
For two intersecting spheres of the same radius and the distance between their centers, the volume of their intersection is given by the formula:
Where:
- (radius of the spheres),
- (distance between the centers of the spheres).
Substituting these values:
Thus, the volume of the solid that lies inside both spheres is cubic units.
Would you like further details or have any questions about this solution?
Here are some related questions:
- How do you find the volume of intersection for spheres with different radii?
- What is the formula for the volume of intersection between two spheres?
- How does the distance between two sphere centers affect the intersection volume?
- How can you find the center and radius of a sphere from its general equation?
- What happens when the distance between two spheres is greater than the sum of their radii?
Tip: When completing the square for sphere equations, remember to group the terms of , , and separately to make the process easier.
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Math Problem Analysis
Mathematical Concepts
Geometry
Spheres
Volume of Intersection of Spheres
Coordinate Geometry
Completing the Square
Formulas
Volume of intersection of two spheres: V = (π (4r - d)^2 (d^2 + 2rd - 3r^2)) / 12d
Theorems
Formula for volume of intersection of two spheres
Suitable Grade Level
Grades 11-12