The minute hand of a clock is 8 cm long. Find the area swept by the minute hand between 8.30 a.m. and 9.05 a.m.

Answer

To find the area swept by the minute hand of a clock between 8:30 a.m. and 9:05 a.m., we can follow these steps:

### Step 1: Determine the length of the minute hand
The length of the minute hand is given as 8 cm.

### Step 2: Calculate the time difference
The time difference between 8:30 a.m. and 9:05 a.m. is:
\[ 9:05 - 8:30 = 35 \text{ minutes} \]

### Step 3: Find the angle swept by the minute hand
The minute hand sweeps 360 degrees in 60 minutes. Therefore, the angle swept per minute is:
\[ \text{Angle per minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees/minute} \]
Now, for 35 minutes, the total angle swept (θ) is:
\[ \theta = 6 \text{ degrees/minute} \times 35 \text{ minutes} = 210 \text{ degrees} \]

### Step 4: Use the formula for the area of a sector
The area (A) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\pi r^2 \theta}{360} \]
Where:
- \( r \) is the radius (length of the minute hand),
- \( \theta \) is the angle in degrees.

Substituting the values:
- \( r = 8 \text{ cm} \)
- \( \theta = 210 \text{ degrees} \)

### Step 5: Substitute the values into the formula
\[ A = \frac{\pi \times (8)^2 \times 210}{360} \]
\[ = \frac{\pi \times 64 \times 210}{360} \]

### Step 6: Simplify the expression
First, we can simplify \( \frac{210}{360} \):
\[ \frac{210}{360} = \frac{7}{12} \]
Now substituting back:
\[ A = \pi \times 64 \times \frac{7}{12} \]
\[ = \frac{64 \times 7 \pi}{12} \]
\[ = \frac{448 \pi}{12} \]
\[ = \frac{112 \pi}{3} \]

### Step 7: Substitute the value of π (approximately 3.14)
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{112 \times \frac{22}{7}}{3} \]
\[ = \frac{2464}{21} \]
Calculating this gives approximately:
\[ A \approx 117.33 \text{ cm}^2 \]

### Final Answer
The area swept by the minute hand between 8:30 a.m. and 9:05 a.m. is approximately:
\[ \frac{448}{21} \text{ cm}^2 \text{ or } 117.33 \text{ cm}^2 \]

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