The volume of a solid sphere is `"4851 m"^(3)` . What is the surface area of the sphere?(Take `pi=(22)/(7)` )

Options

A. `1386m^(2)`
B. `1364m^(2)`
C. `1260m^(2)`
D. `1408m^(2)`

Answer

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

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Set Up the Equation**:
We know the volume of the sphere is \( 4851 \, m^3 \). Therefore, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 4851 \]

3. **Substitute the Value of \( \pi \)**:
We are given \( \pi = \frac{22}{7} \). Substitute this value into the equation:
\[ \frac{4}{3} \times \frac{22}{7} \times r^3 = 4851 \]

4. **Simplify the Equation**:
Multiply both sides by \( \frac{3}{4} \):
\[ \frac{22}{7} r^3 = 4851 \times \frac{3}{4} \]
Calculate \( 4851 \times \frac{3}{4} \):
\[ 4851 \times \frac{3}{4} = \frac{14553}{4} = 3638.25 \]
Thus, we have:
\[ \frac{22}{7} r^3 = 3638.25 \]

5. **Multiply Both Sides by \( \frac{7}{22} \)**:
To isolate \( r^3 \), multiply both sides by \( \frac{7}{22} \):
\[ r^3 = 3638.25 \times \frac{7}{22} \]
Calculate:
\[ r^3 = \frac{3638.25 \times 7}{22} = \frac{25467.75}{22} = 1153.125 \]

6. **Find the Cube Root**:
To find \( r \), take the cube root of \( 1153.125 \):
\[ r = \sqrt[3]{1153.125} \approx 10.5 \, m \]

7. **Calculate the Surface Area**:
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Substitute \( r \) and \( \pi \):
\[ A = 4 \times \frac{22}{7} \times (10.5)^2 \]
Calculate \( (10.5)^2 = 110.25 \):
\[ A = 4 \times \frac{22}{7} \times 110.25 \]
Simplifying:
\[ A = \frac{88 \times 110.25}{7} = \frac{9682}{7} \approx 1383.14 \, m^2 \]

8. **Final Answer**:
The surface area of the sphere is approximately \( 1386 \, m^2 \).