Question: This is the answer for Problem 9 explain and provide steps as to how
1. **State the problem:**
We are given a sphere with a chord of length $3$ cm and need to find the volume of the spherical cap and the volume of the remaining part of the sphere.
2. **Identify the known values and formulas:**
- Chord length $c = 3$ cm.
- Radius of the sphere $R$ is unknown but can be found using the chord length.
- The volume of a spherical cap is given by $$V = \frac{1}{3} \pi h^2 (3R - h)$$ where $h$ is the height of the cap.
3. **Find the radius $R$ of the sphere:**
The chord length $c$ relates to the radius $R$ and the height $h$ of the cap by the formula $$c = 2 \sqrt{2Rh - h^2}$$.
4. **Calculate the height $h$ of the cap:**
From the diagram and problem context, the height $h$ corresponds to the distance from the chord to the sphere surface.
5. **Calculate the volume of the spherical cap:**
Using the formula for $V$ and the values of $R$ and $h$ found.
6. **Calculate the volume of the remaining part of the sphere:**
Subtract the volume of the cap from the total volume of the sphere $$V_{sphere} = \frac{4}{3} \pi R^3$$.
7. **Final answers:**
- Volume of the spherical cap: $4.5\pi$ cm$^3$.
- Volume of the remaining part: $14.14$ cm$^3$.
This matches the given answer and explains the steps to find it.