The volume of a hemisphere is `89.83 cm^(3)`. Find its diameter (in cm.).

Options

A. `3.5`
B. 7
C. 14
D. `10.5`

Answer

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

### Step 1: Understand the volume formula for 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: Relate radius to diameter
The diameter \( D \) is related to the radius by the formula:
\[ D = 2r \]
Thus, we can express the radius in terms of the diameter:
\[ r = \frac{D}{2} \]

### Step 3: Substitute radius in the volume formula
Substituting \( r = \frac{D}{2} \) into the volume formula gives:
\[ V = \frac{2}{3} \pi \left(\frac{D}{2}\right)^3 \]
Simplifying this, we have:
\[ V = \frac{2}{3} \pi \left(\frac{D^3}{8}\right) = \frac{\pi D^3}{12} \]

### Step 4: Set up the equation with the given volume
We know from the problem that the volume \( V = 89.83 \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ 89.83 = \frac{\pi D^3}{12} \]

### Step 5: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we substitute this into the equation:
\[ 89.83 = \frac{\frac{22}{7} D^3}{12} \]

### Step 6: Solve for \( D^3 \)
Rearranging the equation to solve for \( D^3 \):
\[ D^3 = \frac{89.83 \times 12 \times 7}{22} \]
Calculating the right side:
\[ D^3 = \frac{89.83 \times 84}{22} \approx \frac{7545.72}{22} \approx 343.0 \]

### Step 7: Calculate the diameter \( D \)
Now, we take the cube root of \( D^3 \):
\[ D = \sqrt[3]{343} = 7 \, \text{cm} \]

### Final Answer
The diameter of the hemisphere is \( 7 \, \text{cm} \).

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