The total surface area of a metallic hemisphere is `1848 cm^(2)`. The hemisphere is metled to form a solid right circular cone. If the radius of the base of the cone is the same as the radius of the hemisphere, its height is

Options

A. 42 cm
B. 26 cm
C. 28 cm
D. 30 cm

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Find the radius of the 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. We know the TSA is \( 1848 \, \text{cm}^2 \).

Setting up the equation:
\[ 3\pi r^2 = 1848 \]

### Step 2: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[ 3 \times \frac{22}{7} \times r^2 = 1848 \]

### Step 3: Simplify the equation
Multiply both sides by \( 7 \) to eliminate the fraction:
\[ 3 \times 22 \times r^2 = 1848 \times 7 \]
Calculating \( 1848 \times 7 \):
\[ 1848 \times 7 = 12936 \]
So we have:
\[ 66r^2 = 12936 \]

### Step 4: Solve for \( r^2 \)
Divide both sides by \( 66 \):
\[ r^2 = \frac{12936}{66} \]
Calculating \( \frac{12936}{66} \):
\[ r^2 = 196 \]

### Step 5: Find \( r \)
Taking the square root:
\[ r = \sqrt{196} = 14 \, \text{cm} \]

### Step 6: Find the volume of the hemisphere
The volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Substituting \( r = 14 \):
\[ V = \frac{2}{3} \times \frac{22}{7} \times (14)^3 \]

### Step 7: Calculate \( (14)^3 \)
Calculating \( (14)^3 \):
\[ 14^3 = 2744 \]
So,
\[ V = \frac{2}{3} \times \frac{22}{7} \times 2744 \]

### Step 8: Simplify the volume expression
Calculating \( \frac{2 \times 22 \times 2744}{3 \times 7} \):
\[ V = \frac{120736}{21} \]

### Step 9: Find the volume of the cone
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r = 14 \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times (14)^2 \times h \]
Calculating \( (14)^2 \):
\[ (14)^2 = 196 \]
So,
\[ V = \frac{1}{3} \times \frac{22}{7} \times 196 \times h \]

### Step 10: Set the volumes equal
Since the volume of the hemisphere is equal to the volume of the cone:
\[ \frac{120736}{21} = \frac{1}{3} \times \frac{22}{7} \times 196 \times h \]

### Step 11: Solve for \( h \)
Cross-multiplying to solve for \( h \):
\[ 120736 \times 3 \times 7 = 22 \times 196 \times h \]
Calculating \( 22 \times 196 \):
\[ 22 \times 196 = 4312 \]
So,
\[ 120736 \times 21 = 4312h \]
Now, divide both sides by \( 4312 \):
\[ h = \frac{120736 \times 21}{4312} \]

### Step 12: Calculate \( h \)
Calculating \( h \):
\[ h = 28 \, \text{cm} \]

Thus, the height of the cone is \( 28 \, \text{cm} \).

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