What is the measure of central angle of the arc whose length is 11cm and radius of the circle is 14 cm ?

Options

A. `45^(@)`
B. `60^(@)`
C. `75^(@)`
D. `90^(@)`

Answer

To find the measure of the central angle of the arc, we can use the formula for the length of an arc:

\[ \text{Length of arc} = \frac{\theta}{360} \times 2\pi r \]

Where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the circle.

Given:
- Length of arc = 11 cm
- Radius \(r = 14\) cm

### Step 1: Substitute the known values into the formula

We can substitute the values into the formula:

\[ 11 = \frac{\theta}{360} \times 2\pi \times 14 \]

### Step 2: Simplify the equation

First, calculate \(2\pi \times 14\). Using \(\pi \approx \frac{22}{7}\):

\[ 2\pi \times 14 = 2 \times \frac{22}{7} \times 14 = \frac{44 \times 14}{7} = \frac{616}{7} \]

Now, substitute this back into the equation:

\[ 11 = \frac{\theta}{360} \times \frac{616}{7} \]

### Step 3: Multiply both sides by \(360\)

To eliminate the fraction, multiply both sides by \(360\):

\[ 11 \times 360 = \theta \times \frac{616}{7} \]

Calculating \(11 \times 360\):

\[ 3960 = \theta \times \frac{616}{7} \]

### Step 4: Multiply both sides by \(7\)

To get rid of the fraction on the right side, multiply both sides by \(7\):

\[ 3960 \times 7 = \theta \times 616 \]

Calculating \(3960 \times 7\):

\[ 27720 = \theta \times 616 \]

### Step 5: Solve for \(\theta\)

Now, divide both sides by \(616\):

\[ \theta = \frac{27720}{616} \]

Calculating this gives:

\[ \theta = 45 \]

### Final Answer

The measure of the central angle \(\theta\) is \(45\) degrees.

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