Question

A minute hand of a clock is 3 cm long, the distance moved in 20 minutes is:
(a) 3 cm
(b) 9 cm
\[\left( c \right)\dfrac{22}{7}cm\]
\[\left( d \right)\dfrac{44}{7}cm\]

Answer 1

Hint: First draw a rough diagram of a sector of a circle with the minute hand as the radius. Then find the angle made by the minute hand when it moves 20 minutes. Now, convert this angle in radians by using the formula: \[\text{Angle in degrees}=\left( \dfrac{\pi }{180} \right)\] angle in radian. Once the angle is determined, then we will apply the formula, \[\theta =\dfrac{l}{r},\] where \[\theta \] is the angle in radian, ‘r’ is the radius of the sector of the circle and ‘l’ is the length of the arc formed by the movement of the radius, i.e. the minute hand. Now, find the value of ‘l’ which will be our answer.

Complete step by step answer:
We have been provided with a clock having the minute hand of length 3 cm. Now, when it will move then it will form some angle. Let us draw a rough diagram for this.

In the above diagram, we have considered OA as the initial position of the minute hand and after 20 minutes it moves to the position OB. Here, \[\theta \] is the angle subtended by the minute hand during the duration of 20 minutes, r is the radius of the sector of the circle formed or the length of the minute hand and l is the length of the arc, i.e. the distance travelled or moved.
Now, we know that when the minute moves 60 minutes, it forms a complete angle, i.e. 360 degrees. So, applying the unitary method, we get, the angle formed when the minute hand moves 20 minutes,
\[=\dfrac{{{360}^{\circ }}}{{{60}^{\circ }}}\times 20\]
\[={{120}^{\circ }}\]
Now apply the formula for the conversion of angle in degrees into an angle in radians, we get,
\[\Rightarrow {{180}^{\circ }}=\pi \text{ radian}\]
\[\Rightarrow {{120}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\times \text{120 radian}\]
\[\Rightarrow {{120}^{\circ }}=\dfrac{2\pi }{3}\text{radian}=\theta \]
Now, we know that,
\[\Rightarrow \theta =\dfrac{l}{r}\]
Therefore, substituting the values of \[\theta =\dfrac{2\pi }{3}\] and r = 3, we get, the distance moved
\[=l\]
\[=r\theta \]
\[=3\times \dfrac{2\pi }{3}\]
\[=2\pi \text{ }cm\]
Substituting the value of \[\pi =\dfrac{22}{7},\] we get,
\[\Rightarrow l=2\times \dfrac{22}{7}cm\]
\[\Rightarrow l=\dfrac{44}{7}cm\]

So, the correct answer is “Option d”.

Note: One may note that, \['\pi '\] is the real number in the formula whose value is \[\dfrac{22}{7}.\] You must be very careful while applying the formula \[\theta =\dfrac{l}{r}.\] Here, \['\theta '\] must be in radian otherwise if we will consider it in degrees, then we will get a wrong answer. One must remember the conversion formula for this. If we are considering the angle in degrees then we will have to use the formula, \[l=2\pi r\times \dfrac{\theta }{360},\] for the calculation of the length of the arc.