The volume of a sphere is 4.5 `picm^(3)`. Then, its diameter is ……… cm.

Options

A. 3
B. 2
C. 1.5
D. 4

Answer

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

### Step-by-Step Solution

1. **Write down the formula for the volume of a sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Set the volume equal to the given value**:
We know the volume of the sphere is \( 4.5 \pi \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 4.5 \pi \]

3. **Cancel \( \pi \) from both sides**:
Since \( \pi \) is present on both sides of the equation, we can divide both sides by \( \pi \):
\[ \frac{4}{3} r^3 = 4.5 \]

4. **Multiply both sides by \( \frac{3}{4} \)**:
To isolate \( r^3 \), multiply both sides by \( \frac{3}{4} \):
\[ r^3 = 4.5 \times \frac{3}{4} \]

5. **Calculate the right side**:
First, convert \( 4.5 \) to a fraction:
\[ 4.5 = \frac{45}{10} \]
Now, multiply:
\[ r^3 = \frac{45}{10} \times \frac{3}{4} = \frac{135}{40} \]

6. **Simplify \( \frac{135}{40} \)**:
Divide both the numerator and the denominator by 5:
\[ r^3 = \frac{27}{8} \]

7. **Find the cube root of both sides**:
To find \( r \), take the cube root:
\[ r = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} = 1.5 \, \text{cm} \]

8. **Calculate the diameter**:
The diameter \( d \) of the sphere is twice the radius:
\[ d = 2r = 2 \times 1.5 = 3 \, \text{cm} \]

### Final Answer
The diameter of the sphere is **3 cm**.