the curved surface area of a cylindrical pillar is 264 m square and its volume is 924 m cube . Find ratio of the diameter and the height of the pillar.

Options

A. `7:6`
B. `6:7`
C. `3:7`
D. `7:3`

Answer

To solve the problem step by step, we will use the formulas for the curved surface area and volume of a cylinder.

### Step 1: Write down the formulas
The formulas we need are:
- Curved Surface Area (CSA) of a cylinder: \( CSA = 2 \pi r h \)
- Volume (V) of a cylinder: \( V = \pi r^2 h \)

### Step 2: Substitute the given values
From the problem, we know:
- Curved Surface Area (CSA) = 264 m²
- Volume (V) = 924 m³

### Step 3: Set up the equations
Using the formulas:
1. \( 2 \pi r h = 264 \) (1)
2. \( \pi r^2 h = 924 \) (2)

### Step 4: Solve for height (h) in terms of radius (r)
From equation (1):
\[ h = \frac{264}{2 \pi r} = \frac{132}{\pi r} \]

### Step 5: Substitute h into the volume equation
Substituting \( h \) from equation (1) into equation (2):
\[ \pi r^2 \left(\frac{132}{\pi r}\right) = 924 \]
This simplifies to:
\[ 132r = 924 \]

### Step 6: Solve for radius (r)
Now, divide both sides by 132:
\[ r = \frac{924}{132} = 7 \text{ m} \]

### Step 7: Find height (h)
Now substitute \( r \) back into the equation for height:
\[ h = \frac{132}{\pi r} = \frac{132}{\pi \cdot 7} \]
Using \( \pi \approx \frac{22}{7} \):
\[ h = \frac{132}{\frac{22}{7} \cdot 7} = \frac{132}{22} = 6 \text{ m} \]

### Step 8: Calculate the diameter
Diameter \( d \) is given by:
\[ d = 2r = 2 \cdot 7 = 14 \text{ m} \]

### Step 9: Find the ratio of diameter to height
The ratio of diameter to height is:
\[ \text{Ratio} = \frac{d}{h} = \frac{14}{6} = \frac{7}{3} \]

### Final Answer
Thus, the ratio of the diameter to the height of the pillar is \( 7:3 \).

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