Question

The short and long hands of the clock are 4 cm and 6 cm long respectfully. Find the distance travelled by their tips in 8 hours.

Answer 1

Hint: First, calculate the value of angle made by the long hand of the clock in degree and also calculate the angle made by the short hand of the clock and then find the distance travelled by their tips using the formula of the length of the arc of the sector that is $l=\dfrac{\theta }{360{}^\circ }\times 2\pi r$.

Complete step-by-step answer:
Since, the long hand of the clock is the minute hand, and we know that the minute hand travels 1 revolution in 1 hour, that is its angle covered is $360{}^\circ$ in one minute.
So, the long hand of the clock will travel 8 revolutions in 8 hours.
Hence, the distance travelled by long hand of the clock is given by formula of the length of the arc of the sector, that is
$l=\dfrac{\theta }{360{}^\circ }\times 2\pi r$, where r will be the length of the long hand (i.e. minute hand) of the clock.
and $\theta$ is the total angle traversed by the minute hand.
Hence, total angle traversed in 8 hours is:
$\theta =360{}^\circ \times 8$ and r = 6 cm
$\therefore \theta =2880{}^\circ$
$\therefore l=\dfrac{360{}^\circ \times 8}{360{}^\circ }\times 2\times \pi \times 6$cm
Put, $\pi =3.14$, then we will get:
$\Rightarrow l=301.593cm$
Hence, the distance travelled by the longer hand of the clock is 301.593 cm.
Now, the short hand of the clock is the hour hand, we know that hour hand travel angle of $30{}^\circ$ in 1 hours.
Hence, angle travelled by the short hand in 8 hours is:
$\begin{align}
  & \theta =30{}^\circ \times 8 \\
 & \Rightarrow \theta =240{}^\circ \\
\end{align}$ and the length of the short hand is 4 cm.
Now, again by using the formula of the length of the arc of the sector we can find the distance travelled by its tips.
Since, $l=\dfrac{\theta }{360{}^\circ }\times 2\pi r$, here $\theta =240{}^\circ$ and r = length of the short hand = 4 cm.
$\therefore l=\dfrac{240{}^\circ }{360{}^\circ }\times 2\times \pi \times 4cm$
$\Rightarrow l=\dfrac{2}{3}\times 2\times \pi \times 4cm$
$\Rightarrow l=\dfrac{16}{3}\pi cm$
$\therefore l=16.755cm$.
Hence, the distance travelled by the short hand and the long hand of the clock is 16.755 cm and 301.593 cm respectively.

Note: Students are required to take care in calculating the value of $\theta$. There is always the chance of making mistakes. They usually used to calculate the value of $\theta$ only for one revolution even if the clock hand is making more than one revolution.