An arc PQ of a circle substends and angle 1.5 radian at its centre O. Let the radius of the circle is 100cm. If l is arc length `overset(frown)(PQ)` (in cms) and m is area of sector POQ (in `cms^(2))`, then `(m)/(l)` is

Options

A. 50
B. 10
C. 15
D. 20

Answer

To solve the problem step by step, we will calculate the arc length \( l \) and the area of sector \( m \), and then find the ratio \( \frac{m}{l} \).

### Step 1: Calculate the arc length \( l \)
The formula for the length of an arc is given by:
\[ l = r \cdot \theta \]
where \( r \) is the radius and \( \theta \) is the angle in radians.

Given:
- Radius \( r = 100 \) cm
- Angle \( \theta = 1.5 \) radians

Substituting the values:
\[ l = 100 \cdot 1.5 = 150 \text{ cm} \]

### Step 2: Calculate the area of sector \( m \)
The formula for the area of a sector is given by:
\[ m = \frac{1}{2} r^2 \theta \]

Substituting the values:
\[ m = \frac{1}{2} \cdot (100)^2 \cdot 1.5 \]
Calculating \( (100)^2 \):
\[ (100)^2 = 10000 \]
Now substituting this back into the area formula:
\[ m = \frac{1}{2} \cdot 10000 \cdot 1.5 = 5000 \cdot 1.5 = 7500 \text{ cm}^2 \]

### Step 3: Calculate the ratio \( \frac{m}{l} \)
Now we have:
- \( m = 7500 \text{ cm}^2 \)
- \( l = 150 \text{ cm} \)

Calculating the ratio:
\[ \frac{m}{l} = \frac{7500}{150} \]
Simplifying:
\[ \frac{7500}{150} = 50 \]

### Final Answer
Thus, the value of \( \frac{m}{l} \) is \( 50 \).