Question

The hour hand of a clock is 7 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:

Remember these standard rates for clock hands:
- The minute hand rotates at a rate of $6^\circ$ per minute.
- The hour hand rotates at a rate of $0.5^\circ$ per minute.
Keep these constants memorized to quickly solve any clock-related questions.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given that the hour hand of a clock is $7\text{ cm}$ long (which represents the radius, though not needed to find the angle).
We need to determine the angle swept by this hour hand between 7:00 a.m. and 8:10 a.m.

Step 2: Key Formula or Approach:
We need to find the rate of angular rotation of the hour hand of a clock:
1. A clock is divided into 12 hours, which corresponds to a complete rotation of $360^\circ$.
2. Therefore, in 12 hours (or $12 \times 60 = 720\text{ minutes}$), the hour hand sweeps an angle of $360^\circ$.
3. This gives the rate of the hour hand as:
\[ \text{Rate} = \frac{360^\circ}{12\text{ hours}} = 30^\circ\text{ per hour} \]
Or in terms of minutes:
\[ \text{Rate} = \frac{30^\circ}{60\text{ minutes}} = 0.5^\circ\text{ per minute} = \left(\frac{1}{2}\right)^\circ\text{ per minute} \]

Step 3: Detailed Explanation:

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

• 2. Recall the angular rate of movement of the hour hand per minute:
\[ \text{Angle swept in } 1\text{ minute} = \left(\frac{1}{2}\right)^\circ \]

• 3. Multiply the total minutes by the rate per minute to find the total angle swept:
\[ \text{Total Angle Swept} = 70 \times \left(\frac{1}{2}\right)^\circ \]

• 4. Simplify the product:
\[ \text{Total Angle Swept} = 35^\circ \]


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