The volume of a hemisphere is `5749.33 cm^(3)`. Find its diameter (in cm).

Options

A. 14
B. 56
C. 42
D. 28

Answer

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

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

### Step 2: Set the volume equal to the given value.
We know the volume of the hemisphere is \( 5749.33 \, \text{cm}^3 \). Therefore, we can write:
\[ \frac{2}{3} \pi r^3 = 5749.33 \]

### Step 3: Substitute the value of \( \pi \).
Using \( \pi \approx \frac{22}{7} \), we can substitute this into the equation:
\[ \frac{2}{3} \times \frac{22}{7} \times r^3 = 5749.33 \]

### Step 4: Simplify the equation.
To isolate \( r^3 \), we can multiply both sides by \( \frac{3}{2} \) and then by \( \frac{7}{22} \):
\[ r^3 = 5749.33 \times \frac{3}{2} \times \frac{7}{22} \]

### Step 5: Calculate the right-hand side.
Calculating this step-by-step:
1. Calculate \( 5749.33 \times \frac{3}{2} \):
\[ 5749.33 \times 1.5 = 8624.995 \]
2. Now, calculate \( 8624.995 \times \frac{7}{22} \):
\[ 8624.995 \times 0.31818 \approx 2744 \]

Thus, we have:
\[ r^3 = 2744 \]

### Step 6: Find the radius \( r \).
To find \( r \), we take the cube root of \( 2744 \):
\[ r = \sqrt[3]{2744} = 14 \, \text{cm} \]

### Step 7: Calculate the diameter.
The diameter \( d \) of the hemisphere is twice the radius:
\[ d = 2r = 2 \times 14 = 28 \, \text{cm} \]

### Final Answer:
The diameter of the hemisphere is \( 28 \, \text{cm} \).

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