The total surface area of a hemisphere is `4158 cm^2`. Find its diameter (in cm),

Options

A. 21
B. 84
C. 42
D. 63

Answer

To find the diameter of a hemisphere given its total surface area, we can follow these steps:

### Step 1: Understand the formula for the 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 2: Substitute the given total surface area
We know that the total surface area is \( 4158 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 3\pi r^2 = 4158 \]

### Step 3: Substitute the value of \( \pi \)
We can use \( \pi \approx \frac{22}{7} \) for our calculations. Substituting this value into the equation gives:
\[ 3 \cdot \frac{22}{7} \cdot r^2 = 4158 \]

### Step 4: Simplify the equation
Multiply both sides by \( 7 \) to eliminate the fraction:
\[ 3 \cdot 22 \cdot r^2 = 4158 \cdot 7 \]
Calculating \( 4158 \cdot 7 \):
\[ 4158 \cdot 7 = 29106 \]
So, we have:
\[ 66r^2 = 29106 \]

### Step 5: Solve for \( r^2 \)
Now, divide both sides by \( 66 \):
\[ r^2 = \frac{29106}{66} \]
Calculating \( \frac{29106}{66} \):
\[ r^2 = 441 \]

### Step 6: Find the radius \( r \)
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{441} = 21 \, \text{cm} \]

### Step 7: Find the diameter
The diameter \( d \) of the hemisphere is twice the radius:
\[ d = 2r = 2 \times 21 = 42 \, \text{cm} \]

### Final Answer
The diameter of the hemisphere is \( 42 \, \text{cm} \).

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