The volume of a sphere is `38808 cm^(3),` find its diameter and the surface area.

Answer

To find the diameter and surface area of a sphere given its volume, we can follow these steps:

### Step 1: Write down the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Set the volume equal to the given value.
We know the volume of the sphere is \( 38808 \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ 38808 = \frac{4}{3} \pi r^3 \]

### Step 3: Solve for \( r^3 \).
To isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 38808 \times \frac{3}{4\pi} \]
Using \( \pi \approx \frac{22}{7} \), we substitute this value in:
\[ r^3 = 38808 \times \frac{3}{4 \times \frac{22}{7}} = 38808 \times \frac{3 \times 7}{88} \]

### Step 4: Simplify the expression.
Calculating \( \frac{38808 \times 21}{88} \):
1. First, calculate \( 38808 \div 22 = 1764 \).
2. Then, multiply \( 1764 \times 3 = 5292 \).
3. Finally, calculate \( 5292 \times 7 = 37044 \).

So we have:
\[ r^3 = 9261 \]

### Step 5: Find the radius \( r \).
To find \( r \), we take the cube root of \( 9261 \):
\[ r = \sqrt[3]{9261} = 21 \, \text{cm} \]

### Step 6: Calculate the diameter.
The diameter \( d \) of the sphere is given by:
\[ d = 2r = 2 \times 21 = 42 \, \text{cm} \]

### Step 7: Calculate the surface area.
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Substituting the value of \( r \):
\[ A = 4 \times \frac{22}{7} \times (21)^2 \]
Calculating \( (21)^2 = 441 \):
\[ A = 4 \times \frac{22}{7} \times 441 \]
Calculating \( 4 \times 22 = 88 \):
\[ A = \frac{88 \times 441}{7} \]
Calculating \( 441 \div 7 = 63 \):
\[ A = 88 \times 63 = 5544 \, \text{cm}^2 \]

### Final Answers:
- Diameter: \( 42 \, \text{cm} \)
- Surface Area: \( 5544 \, \text{cm}^2 \)

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