In the circle of 5cm.radius, what is the length of the are which subtends and angle of `33^(@)15'` at the centre.

Answer

To find the length of the arc that subtends an angle of \(33^\circ 15'\) at the center of a circle with a radius of 5 cm, we can follow these steps:

### Step 1: Convert the angle into degrees
The given angle is \(33^\circ 15'\). We need to convert the minutes into degrees:
- Since \(1' = \frac{1}{60}^\circ\), we have:
\[ 15' = \frac{15}{60}^\circ = \frac{1}{4}^\circ \]
- Therefore, the total angle in degrees is:
\[ 33^\circ + \frac{1}{4}^\circ = 33.25^\circ = \frac{133}{4}^\circ \]

### Step 2: Use the formula for the length of the arc
The formula for the length of the arc \(L\) is given by:
\[ L = \frac{\theta \times 2\pi r}{360^\circ} \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.

### Step 3: Substitute the values into the formula
Here, \( \theta = \frac{133}{4}^\circ \) and \( r = 5 \) cm. Substituting these values into the formula gives:
\[ L = \frac{\frac{133}{4} \times 2\pi \times 5}{360} \]

### Step 4: Simplify the expression
Calculating the above expression:
\[ L = \frac{133 \times 2\pi \times 5}{4 \times 360} \]
\[ = \frac{133 \times 10\pi}{1440} \]
\[ = \frac{133\pi}{144} \]

### Step 5: Calculate the approximate value
To find the approximate length of the arc, we can use the value of \(\pi \approx 3.14\):
\[ L \approx \frac{133 \times 3.14}{144} \approx 0.923 \pi \text{ cm} \]

### Final Answer
Thus, the length of the arc is:
\[ L \approx 0.923 \pi \text{ cm} \]