Question

The length of a pendulum is 70 cm and it describes an arc of length 88 cm when swings. The angle subtended by the arc at the centre is

• A. \(36^\circ\)
• B. \(70^\circ\)
• C. \(72^\circ\)
• D. \(80^\circ\)

Hint:

To make calculations fast and error-free: Always simplify the numbers before multiplying them out fully.
Cancelling the common factor of 10 from 360 and 440 first makes the calculation extremely simple: \(\theta = \frac{88 \times 36}{44} = 2 \times 36 = 72^\circ\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question relates to the topic of Areas Related to Circles, specifically concerning the length of an arc of a circle.
The swinging pendulum acts as the radius of a circle, and the path traced by its tip forms an arc of a sector.
We are given the length of the pendulum (radius \(r = 70\text{ cm}\)) and the length of the arc described (\(l = 88\text{ cm}\)). We need to find the central angle \(\theta\) subtended by this arc.

Step 2: Key Formula or Approach:
The formula for the length of an arc (\(l\)) of a circle of radius \(r\) with central angle \(\theta\) (in degrees) is: \[ l = \frac{\theta}{360^\circ} \times 2\pi r \] We will substitute the given values into this formula and solve for the unknown angle \(\theta\).

Step 3: Detailed Explanation:
1. State the given values: - Radius of the circle (length of the pendulum), \(r = 70\text{ cm}\)
- Arc length, \(l = 88\text{ cm}\)
- Value of \(\pi = \frac{22}{7}\)
2. Write down the arc length formula: \[ l = \frac{\theta}{360^\circ} \times 2\pi r \] 3. Substitute the values into the formula: \[ 88 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times 70 \] 4. Simplify the right-hand side of the equation: - Divide 70 by 7 to get 10: \[ 88 = \frac{\theta}{360^\circ} \times 2 \times 22 \times 10 \] \[ 88 = \frac{\theta}{360^\circ} \times 440 \] 5. Rearrange the equation to isolate \(\theta\): \[ \theta = \frac{88 \times 360^\circ}{440} \] 6. Simplify the division: - Notice that 440 is divisible by 88: \(440 = 5 \times 88\). Therefore: \[ \theta = \frac{360^\circ}{5} \] \[ \theta = 72^\circ \]

Step 4: Final Answer:
The angle subtended by the arc at the centre is \(72^\circ\).
Therefore, the correct option is (C).