If the volume of a sphere is `65 (10)/(21)cm^(3)`, then its surface area is: (Take `pi= (22)/(7)`)

Options

A. `77 (4)/(7) cm^(2)`
B. `76 (4)/(7)cm^(2)`
C. `78 (4)/(7) cm^(2)`
D. `78 (3)/(7) cm^(2)`

Answer

To find the 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: Substitute the given volume into the formula.
We are given that the volume \( V = 65 \frac{10}{21} \) cm³. First, convert this mixed number into an improper fraction:
\[ V = 65 + \frac{10}{21} = \frac{65 \times 21 + 10}{21} = \frac{1365 + 10}{21} = \frac{1375}{21} \text{ cm}^3 \]

### Step 3: Set up the equation with the volume.
Now, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = \frac{1375}{21} \]

### Step 4: Substitute \( \pi \) with \( \frac{22}{7} \).
Substituting \( \pi \) into the equation:
\[ \frac{4}{3} \cdot \frac{22}{7} \cdot r^3 = \frac{1375}{21} \]

### Step 5: Simplify the equation.
Multiply both sides by \( 3 \) and \( 7 \) to eliminate the fractions:
\[ 4 \cdot 22 \cdot r^3 = \frac{1375 \cdot 3 \cdot 7}{21} \]
This simplifies to:
\[ 88 r^3 = \frac{1375 \cdot 3 \cdot 7}{21} \]

### Step 6: Calculate the right-hand side.
Calculating \( \frac{1375 \cdot 3 \cdot 7}{21} \):
\[ \frac{1375 \cdot 21}{21} = 1375 \]
So we have:
\[ 88 r^3 = 1375 \]

### Step 7: Solve for \( r^3 \).
Now divide both sides by 88:
\[ r^3 = \frac{1375}{88} \]

### Step 8: Calculate \( r \).
To find \( r \), take the cube root:
\[ r = \sqrt[3]{\frac{1375}{88}} \]

### Step 9: Find the surface area.
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Substituting \( \pi \) and \( r \):
\[ A = 4 \cdot \frac{22}{7} \cdot \left(\sqrt[3]{\frac{1375}{88}}\right)^2 \]

### Step 10: Calculate the surface area.
After calculating the above expression, we find:
\[ A = 78 \frac{4}{7} \text{ cm}^2 \]

Thus, the surface area of the sphere is:
\[ \boxed{78 \frac{4}{7} \text{ cm}^2} \]