The minute hand of a clock is 2 cm long. Find the area of the face of the clock described by the minute hand between 7 am and 7 : 15 am.

Answer

To find the area of the face of the clock described by the minute hand between 7 am and 7:15 am, we can follow these steps:

### Step 1: Understand the Movement of the Minute Hand
The minute hand of the clock moves in a circular motion. At 7 am, the minute hand points at the 12, and at 7:15 am, it points at the 3. This means the minute hand covers a quarter of the clock face (90 degrees) from 7 am to 7:15 am.

**Hint:** Remember that the clock is divided into 12 hours, and each hour represents 30 degrees (360 degrees / 12 hours).

### Step 2: Calculate the Angle Covered
The total angle covered by the minute hand from 7 am to 7:15 am is 90 degrees.

**Hint:** To find the angle covered in degrees, consider how many minutes have passed and how many degrees the minute hand moves per minute.

### Step 3: Use the Formula for Area of a Sector
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where:
- \( \theta \) is the angle in degrees,
- \( r \) is the radius of the circle.

**Hint:** The radius in this case is the length of the minute hand.

### Step 4: Substitute the Values into the Formula
Here, the radius \( r = 2 \) cm and the angle \( \theta = 90 \) degrees. Substituting these values into the formula gives:
\[ A = \frac{90}{360} \times \pi \times (2)^2 \]

**Hint:** Simplify \( \frac{90}{360} \) first to make calculations easier.

### Step 5: Simplify the Expression
Calculating the values:
\[ A = \frac{1}{4} \times \pi \times 4 \]
\[ A = \pi \]

**Hint:** Remember that \( \pi \) can be approximated as \( \frac{22}{7} \) for calculations.

### Step 6: Final Calculation
Using \( \pi \approx \frac{22}{7} \):
\[ A \approx \frac{22}{7} \text{ cm}^2 \]

### Final Answer
The area of the face of the clock described by the minute hand between 7 am and 7:15 am is approximately \( \frac{22}{7} \) cm².

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