Question

If the distance travelled by the tip of the minute hand of length $3$ units of a circular clock in $40$ minutes is $k\pi$, then the value of $k$ is

Answer 1

Angle traced by the minute hand in $60min$ is ${360}^{\circ }$
So, the angle traced by the minute hand in $40min$
$=\frac{{360}^{\circ }}{60}\times 40={240}^{\circ }$
We know that,
$\theta =\frac{\text{arc length}}{\text{radius}}\Rightarrow \frac{\pi }{{180}^{\circ }}\times {240}^{\circ }=\frac{\text{arc length}}{3}\Rightarrow \text{arc length}=4\pi$

Therefore, the distance travelled by the minute hand is $4\pi .$
Hence, $k=4.$