Answer:
Step-by-step explanation:
Here's how to find the volume of the ring and the value of r for half the sphere's volume:
1. Volume of the Ring:
Let's denote the volume of the sphere (without the hole) as V_sphere and the volume of the ring as V_ring.
a. Volume of the Sphere:
V_sphere = (4/3)πR^3 (Formula for sphere volume)
Here, R = 6, so V_sphere = (4/3)π (6^3) = 288π
b. Volume of the Removed Cylinder with Hemispheres:
Imagine the removed material as a cylinder with two hemispheres attached at the ends.
i. Volume of Cylinder:
The cylinder's radius is r and its height is equal to the distance between the two flat surfaces created by cutting the sphere (which is 2 times the square root of R^2 - r^2 using the Pythagorean theorem). However, for the ring volume calculation, the height doesn't matter. We'll just denote it as h for now.
Volume of Cylinder = πr^2 * h
ii. Volume of a Hemisphere:
The removed material also includes two hemispheres on both ends of the cylinder. The radius of each hemisphere is also r.
Volume of 1 Hemisphere = (2/3)πr^3
iii. Total Volume Removed:
Total Volume Removed = Volume of Cylinder + 2 * Volume of Hemisphere
= πr^2 * h + 2 * (2/3)πr^3
c. Volume of the Ring:
The volume of the ring is the original sphere's volume minus the volume of the removed material.
V_ring = V_sphere - Total Volume Removed
V_ring = 288π - (πr^2 * h + 2 * (2/3)πr^3)
2. Volume of the Ring as Half the Sphere's Volume:
We want to find the value of r when V_ring = (1/2) * V_sphere
Substitute the expressions for V_ring and V_sphere:
288π - (πr^2 * h + 2 * (2/3)πr^3) = (1/2) * 288π
Simplify and solve for r:
576π - (πr^2 * h + (4/3)πr^3) = 144π
(4/3)πr^3 - πr^2 * h = 432π
Notice that the term with h cancels out because the volume of the ring shouldn't depend on the cylinder's height in this scenario. We are only interested in the radius r.
Therefore:
(4/3)πr^3 - πr^2 * h = 432π becomes (4/3)πr^3 = 432π
Divide both sides by (4/3)π:
r^3 = 324
Take the cube root of both sides (round to two decimal places):
r ≈ 6.24
Answer:
V_ring = 288π - (πr^2 * h + 2 * (2/3)πr^3) (in terms of r)
r ≈ 6.24 (for the ring volume to be half the sphere's volume)
