A solid metal sphere is cut through its centre into two equal parts . Find the total surface area of each part if the radius of the sphere is 7 cm.

Options

A. `462 cm^(2)`
B. `231 cm^(2)`
C. `308 cm^(2)`
D. `115.5 cm^(2)`

Answer

To find the total surface area of each hemisphere after cutting a solid metal sphere through its center, we can follow these steps:

### Step 1: Understand the Problem
We have a solid sphere with a radius of 7 cm. When it is cut through its center, it forms two equal parts, which are hemispheres.

### Step 2: Formula for Total Surface Area of a Hemisphere
The total surface area (TSA) of a hemisphere is given by the formula:
\[ \text{TSA} = 3\pi r^2 \]
where \( r \) is the radius of the hemisphere.

### Step 3: Substitute the Radius
Given that the radius \( r \) is 7 cm, we can substitute this value into the formula:
\[ \text{TSA} = 3\pi (7)^2 \]

### Step 4: Calculate \( r^2 \)
First, calculate \( 7^2 \):
\[ 7^2 = 49 \]

### Step 5: Substitute \( r^2 \) into the Formula
Now substitute \( 49 \) back into the TSA formula:
\[ \text{TSA} = 3\pi \times 49 \]

### Step 6: Calculate \( 3 \times 49 \)
Now calculate \( 3 \times 49 \):
\[ 3 \times 49 = 147 \]

### Step 7: Substitute \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we can now calculate the TSA:
\[ \text{TSA} = 147 \times \frac{22}{7} \]

### Step 8: Simplify the Calculation
To simplify:
\[ \text{TSA} = 147 \times \frac{22}{7} = 147 \div 7 \times 22 \]
Calculating \( 147 \div 7 \):
\[ 147 \div 7 = 21 \]
Now multiply by 22:
\[ 21 \times 22 = 462 \]

### Step 9: Final Answer
Thus, the total surface area of each hemisphere is:
\[ \text{TSA} = 462 \, \text{cm}^2 \]

### Summary
The total surface area of each part (hemisphere) is **462 cm²**.

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