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.

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

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

To find the volume of the resulting solid, we need to calculate the volumes of the sphere and the cylinder separately, and then subtract the volume of the cylinder from the volume of the sphere.

First, let's find the volume of the sphere. The formula for the volume of a sphere is given by:

V_sphere = (4/3) * π * r^3

Given that the radius of the sphere is 12 units, we can substitute this value into the formula:

V_sphere = (4/3) * π * 12^3
= (4/3) * π * 1728
= (4/3) * 3.1416 * 1728

Simplifying:

V_sphere = 9047.84

Next, let's find the volume of the cylinder. The formula for the volume of a cylinder is given by:

V_cylinder = π * r^2 * h

In this case, the radius of the cylinder is 6 units (half of the radius of the sphere). To find the height of the cylinder, we need to subtract twice the radius of the cylinder from the height of the sphere, since the hole goes through the center:

h = 2 * r_sphere - 2 * r_cylinder
= 2 * 12 - 2 * 6
= 24 - 12
= 12 units

Substituting the values into the formula:

V_cylinder = π * 6^2 * 12
= 3.1416 * 36 * 12

Simplifying:

V_cylinder = 1357.168

Finally, to find the volume of the resulting solid, we subtract the volume of the cylinder from the volume of the sphere:

V_resulting_solid = V_sphere - V_cylinder
= 9047.84 - 1357.168

Simplifying:

V_resulting_solid = 7690.672

Therefore, the volume of the resulting solid is approximately 7690.672 cubic units.

To find the volume of the resulting solid, we need to subtract the volume of the hole from the volume of the original ball.

First, let's find the volume of the ball:
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
In this case, the radius of the ball is 12, so the volume of the ball is:
V_ball = (4/3)π(12)^3 ≈ 7238.22947 cubic units.

Next, let's find the volume of the hole:
The shape of the hole is a cylinder with a radius of 6 and a height equal to the diameter of the ball, which is also 12.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
In this case, the volume of the hole is:
V_hole = π(6)^2(12) = 432π cubic units.

Finally, we can find the volume of the resulting solid by subtracting the volume of the hole from the volume of the ball:
V_result = V_ball - V_hole = 7238.22947 - 432π ≈ 5796.71874 cubic units.