Problem 1) Start with a cube that has edges of length 1. Circumscribe a sphere around the cube, and then circumscribe another cube around the sphere. What is the volume of the large cube circumscribed around the sphere?
Use a (x,y,z) Cartesian coordinate system.
Place one corner of the cube at (0,0,0) and one corner at (1,1,1).
The distance from (0,0,0) to (1,1,1) is (12+12+12)1/2=31/2.
31/2 is the length of the following distances.
Main diagonal of the small cube
Diameter of the sphere
Edge length of the large cube
The volume of the large cube is (31/2)3=33/2=3(31/2)
Problem 2) Start with a sphere that has a radius of length 1. Circumscribe a cube around the sphere, and then circumscribe another sphere around the cube. What is the radius of the large sphere circumscribed around the cube?
Use a (x,y,z) Cartesian coordinate system. The radius of the small sphere is 1 so the cube has edges of length 2.
Place the center of the cube at (0,0,0) and one corner of the cube at (1,1,1).
The distance from (0,0,0) to (1,1,1) is (12+12+12)1/2=31/2.
Solutions
Problem 1) Start with a cube that has edges of length 1. Circumscribe a sphere around the cube, and then circumscribe another cube around the sphere. What is the volume of the large cube circumscribed around the sphere?
Use a (x,y,z) Cartesian coordinate system.
Place one corner of the cube at (0,0,0) and one corner at (1,1,1).
The distance from (0,0,0) to (1,1,1) is (12+12+12)1/2=31/2.
31/2 is the length of the following distances.
The volume of the large cube is (31/2)3=33/2=3(31/2)
Problem 2) Start with a sphere that has a radius of length 1. Circumscribe a cube around the sphere, and then circumscribe another sphere around the cube. What is the radius of the large sphere circumscribed around the cube?
Use a (x,y,z) Cartesian coordinate system. The radius of the small sphere is 1 so the cube has edges of length 2.
Place the center of the cube at (0,0,0) and one corner of the cube at (1,1,1).
The distance from (0,0,0) to (1,1,1) is (12+12+12)1/2=31/2.
31/2 is also the radius of the large sphere.