The volume of a hemisphere is `89""5/6cm^(3)`. Find its diameter.

Answer

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

### Step 1: Understand 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: Convert the mixed fraction to an improper fraction.
The volume given is \( 89 \frac{5}{6} \) cm³. We need to convert this mixed fraction into an improper fraction:
\[ 89 \frac{5}{6} = \frac{89 \times 6 + 5}{6} = \frac{534 + 5}{6} = \frac{539}{6} \]

### Step 3: Substitute the volume into the volume formula.
Now, we substitute the volume into the formula:
\[ \frac{2}{3} \pi r^3 = \frac{539}{6} \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{2}{3} \times \frac{22}{7} \times r^3 = \frac{539}{6} \]

### Step 4: Simplify the equation.
Multiply both sides by \( 3 \) to eliminate the fraction:
\[ 2 \times \frac{22}{7} \times r^3 = \frac{539 \times 3}{6} \]
Now simplify the right side:
\[ \frac{539 \times 3}{6} = \frac{1617}{6} \]

### Step 5: Solve for \( r^3 \).
Now, rearranging gives:
\[ r^3 = \frac{1617 \times 7}{2 \times 22} \]
Calculating the right side:
\[ r^3 = \frac{1617 \times 7}{44} \]
Now calculate \( 1617 \div 11 = 147 \) (since both 1617 and 44 are divisible by 11):
\[ r^3 = \frac{147 \times 7}{4} \]
Calculating \( 147 \times 7 = 1029 \):
\[ r^3 = \frac{1029}{4} \]

### Step 6: Find the value of \( r \).
Now, we need to take the cube root:
\[ r = \sqrt[3]{\frac{1029}{4}} \]
Calculating \( \frac{1029}{4} = 257.25 \):
\[ r \approx 6.349 \text{ cm} \]

### Step 7: Calculate the diameter.
The diameter \( d \) is given by:
\[ d = 2r \approx 2 \times 6.349 \approx 12.698 \text{ cm} \]

### Final Answer:
The diameter of the hemisphere is approximately \( 12.7 \) cm.