Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm. `(" Use "pi =(22)/(7))`

Options

A. `11^(@)36`
B. `12^(@)36`
C. `12^(@)30`
D. `12^(@)34`

Answer

To find the degree measure of the angle subtended at the center of a circle by an arc of length 22 cm, we can follow these steps:

### Step 1: Identify the given values
- Length of the arc (l) = 22 cm
- Radius of the circle (r) = 100 cm

### Step 2: Use the formula to find the angle in radians
The formula to find the angle (θ) in radians subtended by an arc is given by:
\[ \theta = \frac{l}{r} \]
Substituting the values we have:
\[ \theta = \frac{22}{100} \]

### Step 3: Simplify the fraction
Simplifying \(\frac{22}{100}\):
\[ \theta = \frac{11}{50} \text{ radians} \]

### Step 4: Convert radians to degrees
To convert radians to degrees, we use the conversion factor \(\frac{180}{\pi}\):
\[ \text{Degrees} = \theta \times \frac{180}{\pi} \]
Substituting the value of \(\theta\):
\[ \text{Degrees} = \frac{11}{50} \times \frac{180}{\frac{22}{7}} \]

### Step 5: Simplify the conversion
This can be simplified as follows:
\[ \text{Degrees} = \frac{11 \times 180 \times 7}{50 \times 22} \]

### Step 6: Calculate the numerator and denominator
Calculating the numerator:
\[ 11 \times 180 \times 7 = 13860 \]
Calculating the denominator:
\[ 50 \times 22 = 1100 \]

### Step 7: Divide to get the degree measure
Now divide the numerator by the denominator:
\[ \text{Degrees} = \frac{13860}{1100} = 12.6 \]

### Step 8: Convert the decimal to degrees and minutes
The whole number part is 12 degrees. To convert the decimal part (0.6) to minutes:
\[ 0.6 \times 60 = 36 \text{ minutes} \]
Thus, the angle in degrees and minutes is:
\[ 12 \text{ degrees } 36 \text{ minutes} \]

### Final Answer
The degree measure of the angle subtended at the center of the circle by the arc of length 22 cm is \(12^\circ 36'\).

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