Question

A ball of radius 12 has a round hole of radius 6 drilled through its center. Find the volume of the resulting solid.

I tried finding the volume of the sphere and the volume of the cyclinder then subtract however that did not work.

Answer 1

the part that is cut out,what you call a "cylinder" is not really at cylinder.
you are forgetting about the caps on each end of your 'cylinder'

we will have to use Calculus to do that
Visualize a circle, centre at the origin and radius of 12,rotating about the x-axis resulting in our sphere.

NOw visualize a drill bit of radius 3 as the x-axis, drilling out a hole.

volume of sphere = (4/3)pi(12)^3 = 7238.229
(you probably got that)


now the 'cylinder will cut at (√135,3)and (-√135,3)
so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 1642.59

( I am going to assume you got an answer of 7238.229-1642.59 = 5595.639)

I will calculate one of the "caps", then subtract twice that from the above answer.
vol. of cap = pi[integral](144-x^2)dx from √135 to 12
= pi[144x - (1/3)x^3│ from √135 to 12
= 5.4159
CHECK MY ARITHMETIC, THIS IS WHERE I USUALLY SCREW UP

so total volume
= 7238.229 - 1642.59 - 2(5.4159
= 5584.8072

Answer 2

ARRGGGHH! ARITHMETIC ERROR!!

<< so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 1642.59 >>

should have said:
so the volume of the cylinder with flat tops = pi(3)^2(2(√135)) = 657.036

and then

<<( I am going to assume you got an answer of 7238.229-1642.59 = 5595.639) >>

should say:

( I am going to assume you got an answer of 7238.229-657.036 = 6581.19

and finally at the end
<< so total volume
= 7238.229 - 1642.59 - 2(5.4159
= 5584.8072 >>

should say:

so total volume
= 7238.229 - 657.036 - 2(5.4159
= 6570.36