Calculating Sector Area Swept by Clock Hand

The problem asks for the area swept by the minute hand of a wall clock over a specific time interval. The minute hand's length acts as the radius of the sector it sweeps.

Key Information Extraction

• Length of the minute hand (Radius, $r$): $6\text{ cm}$
• Time interval: 5 minutes

Determining the Angle Swept

A clock face represents a full circle ($360^\circ$). The minute hand completes a full circle ($360^\circ$) in 60 minutes.

• Angle swept per minute = $\frac{360^\circ}{60} = 6^\circ$ per minute.
• Angle swept in 5 minutes = $5 \times 6^\circ = 30^\circ$.

To use the sector area formula, convert the angle to radians:

• Angle in radians, $\theta$ = $30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}$ radians.

Calculating the Area of the Sector

The area of a sector is given by the formula $A = \frac{1}{2} r^2 \theta$, where $r$ is the radius and $\theta$ is the angle in radians.

• Substitute the known values: $r = 6\text{ cm}$ and $\theta = \frac{\pi}{6}$.
• Area, $A$ = $\frac{1}{2} \times (6\text{ cm})^2 \times \frac{\pi}{6}$
• Area, $A$ = $\frac{1}{2} \times 36\text{ cm}^2 \times \frac{\pi}{6}$
• Area, $A$ = $18\text{ cm}^2 \times \frac{\pi}{6}$
• Area, $A$ = $3\pi\text{ cm}^2$.

Numerical Calculation and Final Answer

Using the approximate value of $\pi \approx 3.14159$:

• Area, $A$ $\approx 3 \times 3.14159 \text{ cm}^2$
• Area, $A$ $\approx 9.42477 \text{ cm}^2$.

Rounding to two decimal places gives $9.42\text{ cm}^2$. Comparing this to the options, $9.43\text{ cm}^2$ is the closest value.