Question

The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (use \[\pi =3.14\])

Answer 1

Hint: The minute hand of a clock will complete one complete circle in one hour. So it makes \[{{360}^{\circ }}\]in one hour. In one minute the minute hand of a watch is displaced by 6 degrees. So it is displaced by an angle 240 degrees in 40 minutes. We will get the distance moved by minute hand of a watch by formula \[l=r\theta \].

Complete step-by-step answer:
Given r= length of minute hand of a watch =1.5cm
Now we have to find \[\theta \]
We know that minute hand makes one revolution in one hour or else we can say that minute hand completes \[{{360}^{\circ }}\]in one hour
Degrees completed in 60 minutes = \[{{360}^{\circ }}\]
Degrees completed in 1 minute= \[\dfrac{{{360}^{\circ }}}{{{60}^{\circ }}}={{6}^{\circ }}\]
Degrees completed in 40 minutes = \[{{6}^{\circ }}\times 40={{240}^{\circ }}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
So we obtained that \[\theta ={{240}^{\circ }}\]now we should convert them into degrees
\[={{240}^{\circ }}\times \dfrac{\pi }{180}\]
\[=\dfrac{4\pi }{3}\]
The angle we got in radians \[=\dfrac{4\pi }{3}\]radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2)
Now put the values of l, r in the above formula that is \[l=r\theta \]
\[\begin{align}
  & =1.5\times \dfrac{4\pi }{3} \\
 & =0.5\times 4\pi \\
\end{align}\]
\[=2\pi \]
Given that \[\pi =3.14\]
\[\begin{align}
  & =2\times 3.14 \\
 & =6.28cm \\
\end{align}\]
So, the minutes hand tip will move 6.28cm in 40 minutes

Note: The length of the arc is given by formula \[l=r\theta \]in this \[\theta \]is the angle subtended and it is in radians but in the problem \[\theta \]is given in degrees so we have to convert to radians. To convert degrees to radians we have to multiply the given degrees with \[\dfrac{\pi }{180}\]then we will get angle subtended in radians.