Find the volume of the solid that lies inside both of the spheres x^2+y^2

Aneeka Hunt

Aneeka Hunt

Answered question

2021-10-28

Determine the solid's volume inside each of the two spheres
x2+y2+z2+4x2y+4z+5=0 
and 
x2+y2+z2=4

Answer & Explanation

wornoutwomanC

wornoutwomanC

Skilled2021-10-29Added 81 answers

Step1 
x2+y2+z2+4x2y+4z+5=0 
(x2+4x)+(y22y)+(z2+4z)+5=0 
(x2+4x+44)+(y22y+11)+(z2+4z+44)+5=0 
(x2+4x+4)4+(y22y+1)1+(z2+4z+4)4+5=0 
(x+2)2+(y1)2+(z+2)2=4 
This sphere has a radius of 2 and a center at (-2,1,-2)
Step 2 
I am going to use the following formula : 
V=π12(4R+d)(2Rd)2 
where d is the separation between the centers of two spheres
And R is the radius of the two spheres, that is 2 in this case 
Link to the proof of this formula is given in comment section. 
Step 3 
Distance between the two centres is 
d=(2)2+1(1)2+(2)2=4+1+4=9=3 
Step 4 
Substitute d=3 and R=2, to get the volume 
V=π12(42+3)(223)22.88 
Result 
V2.88

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?