Question

The hour hand of a clock is $7\text{ cm}$ long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :

• A. $\left(\frac{35}{4}\right)^\circ$
• B. $\left(\frac{35}{2}\right)^\circ$
• C. $35^\circ$
• D. $70^\circ$

Hint:

A useful thumb rule for clocks:
The hour hand moves at a rate of $30^\circ$ per hour, or $0.5^\circ$ per minute.
The minute hand moves at a rate of $360^\circ$ per hour, or $6^\circ$ per minute.
For 1 hour and 10 minutes, the hour hand moves:
\[ (1 \times 30^\circ) + (10 \times 0.5^\circ) = 30^\circ + 5^\circ = 35^\circ \]
This breakdown makes mental calculation extremely quick.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic is clock-based problems in mathematics, which involves calculating angles swept by the hands of a clock over a specific time interval.
We are given the length of the hour hand (which represents the radius of the circular path, though it is extra information not needed to find the angle itself).
We need to calculate the angle swept by the hour hand from 7:00 a.m. to 8:10 a.m.

Step 2: Key Formula or Approach:
We must determine the angular speed of the hour hand of a clock:

• The hour hand completes one full rotation of $360^\circ$ in exactly 12 hours.

• Total minutes in 12 hours = $12 \times 60 = 720\text{ minutes}$.

• Therefore, the angle swept by the hour hand in 1 minute is:
\[ \text{Angular speed} = \frac{360^\circ}{720\text{ minutes}} = 0.5^\circ\text{ per minute} = \left(\frac{1}{2}\right)^\circ\text{ per minute} \]

Once we have this rate, we calculate the total duration in minutes and multiply by the rate.

Step 3: Detailed Explanation:

• Calculate the total time elapsed between 7:00 a.m. and 8:10 a.m.:
From 7:00 a.m. to 8:00 a.m. is 1 hour ($60\text{ minutes}$).
From 8:00 a.m. to 8:10 a.m. is 10 minutes.
Total duration = $60\text{ minutes} + 10\text{ minutes} = 70\text{ minutes}$.

• Determine the angle swept per minute by the hour hand:
Since the hour hand covers $360^\circ$ in 12 hours:
Angle swept in 1 hour = $\frac{360^\circ}{12} = 30^\circ$
Since 1 hour = 60 minutes, the angle swept in 1 minute is:
\[ \frac{30^\circ}{60} = 0.5^\circ = \left(\frac{1}{2}\right)^\circ \]

• Compute the angle swept in 70 minutes:
\[ \text{Angle swept} = 70 \times 0.5^\circ \]
\[ \text{Angle swept} = 35^\circ \]


Step 4: Final Answer:
The angle swept by the hour hand is $35^\circ$, which corresponds to option (C).