(i) The numberical value of the volume and surface of a sphere are equal. Find the diameter of the sphere.
(ii) The curved surved surface of a sphere is equal to the area of a circle of radius 2.8 cm. Find the volume of the sphere.

Answer

Let's solve the problem step by step.

### Part (i): Finding the Diameter of the Sphere

1. **Understanding the Problem**: We need to find the diameter of a sphere where the numerical values of its volume and surface area are equal.

2. **Formulas**:
- Volume of a sphere: \( V = \frac{4}{3} \pi r^3 \)
- Surface area of a sphere: \( A = 4 \pi r^2 \)

3. **Setting the Equations Equal**: Since the volume and surface area are equal, we can set the two equations equal to each other:
\[ \frac{4}{3} \pi r^3 = 4 \pi r^2 \]

4. **Canceling Common Terms**: We can cancel \( 4 \pi \) from both sides:
\[ \frac{1}{3} r^3 = r^2 \]

5. **Rearranging the Equation**: Multiply both sides by 3 to eliminate the fraction:
\[ r^3 = 3r^2 \]

6. **Factoring the Equation**: Factor out \( r^2 \):
\[ r^2 (r - 3) = 0 \]
This gives us two solutions: \( r^2 = 0 \) or \( r - 3 = 0 \). Since \( r \) cannot be zero, we have:
\[ r = 3 \]

7. **Finding the Diameter**: The diameter \( d \) of the sphere is given by:
\[ d = 2r = 2 \times 3 = 6 \text{ units} \]

### Part (ii): Finding the Volume of the Sphere

1. **Understanding the Problem**: The curved surface area of the sphere is equal to the area of a circle with a radius of 2.8 cm.

2. **Formulas**:
- Curved Surface Area (CSA) of a sphere: \( CSA = 4 \pi r^2 \)
- Area of a circle: \( A = \pi R^2 \) where \( R = 2.8 \text{ cm} \)

3. **Setting the Areas Equal**: According to the problem:
\[ 4 \pi r^2 = \pi (2.8)^2 \]

4. **Canceling Common Terms**: We can cancel \( \pi \) from both sides:
\[ 4r^2 = (2.8)^2 \]

5. **Calculating \( (2.8)^2 \)**:
\[ (2.8)^2 = 7.84 \]
So we have:
\[ 4r^2 = 7.84 \]

6. **Dividing by 4**:
\[ r^2 = \frac{7.84}{4} = 1.96 \]

7. **Finding the Radius**: Taking the square root:
\[ r = \sqrt{1.96} = 1.4 \text{ cm} \]

8. **Finding the Volume**: Now, we can find the volume of the sphere using the volume formula:
\[ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (1.4)^3 \]

9. **Calculating \( (1.4)^3 \)**:
\[ (1.4)^3 = 2.744 \]
Therefore:
\[ V = \frac{4}{3} \pi (2.744) \approx 11.52 \text{ cm}^3 \text{ (using } \pi \approx 3.14\text{)} \]

### Final Answers:
- (i) The diameter of the sphere is **6 units**.
- (ii) The volume of the sphere is approximately **11.52 cm³**.