The volume of a hemisphere is `2425.5 cm^(3)`. Find its diameter (in cm).

Options

A. 10.5
B. 42
C. 21
D. 31.5

Answer

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

### Step 1: Write down the formula for the volume of a hemisphere.
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.

### Step 2: Set the volume equal to the given volume.
We know the volume of the hemisphere is \( 2425.5 \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ \frac{2}{3} \pi r^3 = 2425.5 \]

### Step 3: Substitute the value of \( \pi \).
For simplicity, we will use \( \pi \approx \frac{22}{7} \). Substituting this into the equation gives:
\[ \frac{2}{3} \times \frac{22}{7} \times r^3 = 2425.5 \]

### Step 4: Simplify the equation.
Multiplying both sides by \( \frac{3}{2} \) to isolate \( r^3 \):
\[ \frac{22}{7} r^3 = 2425.5 \times \frac{3}{2} \]
Calculating the right side:
\[ 2425.5 \times \frac{3}{2} = 2425.5 \times 1.5 = 3638.25 \]
So we have:
\[ \frac{22}{7} r^3 = 3638.25 \]

### Step 5: Multiply both sides by \( \frac{7}{22} \).
To solve for \( r^3 \), multiply both sides by \( \frac{7}{22} \):
\[ r^3 = 3638.25 \times \frac{7}{22} \]
Calculating the right side:
\[ r^3 = 3638.25 \times \frac{7}{22} = 3638.25 \times 0.31818 \approx 1155 \]

### Step 6: Take the cube root to find \( r \).
Now we need to find \( r \):
\[ r = \sqrt[3]{1155} \]
Calculating the cube root gives approximately:
\[ r \approx 10.5 \, \text{cm} \]

### Step 7: Find the diameter.
The diameter \( d \) is given by:
\[ d = 2r = 2 \times 10.5 = 21 \, \text{cm} \]

### Final Answer:
The diameter of the hemisphere is \( 21 \, \text{cm} \).

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