The radius of a cylinder & a sphere is same, and ratio of height and radius of cylinder is 2 : 1.If the volume of sphere is `288 pi cm^(3)` then find the volume of cylinder? (in `cm ^(3))`

Options

A. A)`432pi`
B. B)`426 pi`
C. C)`420 pi`
D. D)`444pi`

Answer

To find the volume of the cylinder given the volume of the sphere and the relationship between the height and radius of the cylinder, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the volume of the sphere**:
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
We know from the problem that the volume of the sphere is \(288 \pi \, \text{cm}^3\).

2. **Set up the equation**:
Equate the volume of the sphere to the given volume:
\[ \frac{4}{3} \pi r^3 = 288 \pi \]

3. **Cancel \(\pi\) from both sides**:
\[ \frac{4}{3} r^3 = 288 \]

4. **Multiply both sides by \(\frac{3}{4}\)** to isolate \(r^3\):
\[ r^3 = 288 \times \frac{3}{4} \]
\[ r^3 = 216 \]

5. **Find the radius \(r\)**:
Taking the cube root of both sides gives:
\[ r = \sqrt[3]{216} = 6 \, \text{cm} \]

6. **Determine the height of the cylinder**:
The problem states that the ratio of the height \(h\) to the radius \(r\) of the cylinder is \(2:1\). Thus, we can express height as:
\[ \frac{h}{r} = 2 \implies h = 2r \]
Substituting the value of \(r\):
\[ h = 2 \times 6 = 12 \, \text{cm} \]

7. **Calculate the volume of the cylinder**:
The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values of \(r\) and \(h\):
\[ V = \pi (6^2) (12) \]
\[ V = \pi (36) (12) \]
\[ V = 432 \pi \, \text{cm}^3 \]

### Final Answer:
The volume of the cylinder is \(432 \pi \, \text{cm}^3\).

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