Ishan whishe Irfan 'Good Morning' when the hour hand of a (measured clockwise) clock is positioned between 9 and 10 . The angle between the two hands is `145^(@)` . The time shown by the clock is

Options

A. ` 9 . 0 8 AM`
B. `9 . 10 AM`
C. ` 9 . 12 AM`
D. `9 . 15 AM`

Answer

To find the time shown by the clock when Ishan wished Irfan "Good Morning," we need to determine the position of the hour and minute hands based on the given angle of 145 degrees between them.

### Step-by-Step Solution:

1. **Understanding the Clock Positions:**
The hour hand is positioned between 9 and 10, which means the time is somewhere between 9:00 and 10:00.

2. **Calculating the Movement of the Hands:**
- The minute hand moves 360 degrees in 60 minutes, which means it moves \( \frac{360}{60} = 6 \) degrees per minute.
- The hour hand moves 30 degrees in 60 minutes, which means it moves \( \frac{30}{60} = 0.5 \) degrees per minute.

3. **Initial Angle at 9:00:**
At exactly 9:00, the hour hand is at 9 (which corresponds to 270 degrees) and the minute hand is at 12 (which corresponds to 0 degrees). The angle between them at this time is:
\[ 270 - 0 = 270 \text{ degrees} \]
However, since we measure the angle in the clockwise direction, the angle is actually 90 degrees (the smaller angle).

4. **Finding the Change in Angle Over Time:**
As time passes, the angle between the two hands changes. Each minute, the minute hand moves 6 degrees forward, while the hour hand moves 0.5 degrees forward. Thus, the net increase in the angle between the two hands per minute is:
\[ 6 - 0.5 = 5.5 \text{ degrees} \]

5. **Setting Up the Equation:**
We need to find the time \( x \) (in minutes) after 9:00 when the angle between the hands becomes 145 degrees. Starting from the initial angle of 90 degrees, we can express the angle after \( x \) minutes as:
\[ 90 + 5.5x = 145 \]

6. **Solving for \( x \):**
Rearranging the equation gives:
\[ 5.5x = 145 - 90 \]
\[ 5.5x = 55 \]
\[ x = \frac{55}{5.5} = 10 \]

7. **Calculating the Time:**
Since \( x = 10 \), this means 10 minutes have passed since 9:00. Therefore, the time shown on the clock is:
\[ 9:10 \]

### Final Answer:
The time shown by the clock is **9:10**.

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