Question

The minute hand of a clock is \[2.1\] cm long. How far does its tip move in 20 minutes?

Answer 1

Hint:
As we know that the tip of the minute hand makes a complete circle when 60 minutes get completed. When the circle completes one round then it is of \[2\pi \]degrees. After this, we will convert the degrees into radians for 1 minute and we need to calculate for 20 minutes so, we will multiply the converted from by 20. Next, we will use the formula \[\theta = \dfrac{l}{r}\] and keep it equal to the expression we have found and we will evaluate the value of \[l\].

Complete step by step solution:
Consider the length of the minute hand of a clock that is \[2.1\] cm.
First, we will calculate the distance the minute hand moves in 20 minutes.
We know that when 60 minutes get completed implies that the tip of a minute hand makes a complete circle.
And when the circle makes one round implies that the circle is of \[2\pi \] degrees.
Thus, we get that,
\[
   \Rightarrow 2\pi = 60 \\
   \Rightarrow \pi = 30 \\
 \]
Now, we will find how much radian move in 20 minutes.
Since 1minute equals to \[\dfrac{\pi }{{30}}\] in radians
Therefore, we will calculate the radian for 20 minutes,
Thus, we get,
\[ \Rightarrow \theta = 20 \times \dfrac{\pi }{{30}} = \dfrac{{2\pi }}{3}\]
\[ \Rightarrow \theta = \dfrac{{2\pi }}{3}\] ----(1)
Next, we are given in the question that the length of the minute hand clock is \[2.1\] cm which gives us the value of the radius of the clock.
Also, the distance covered by the tip of the minute is given by \[l\].
Thus, we get the formula as \[\theta = \dfrac{l}{r}\]
Substituting the value of \[r\] in the above formula, we get,
\[ \Rightarrow \theta = \dfrac{l}{{2.1}}\] ---(2)
Now, we will compare equation (1) and (2) to find the value of \[l\].
Thus, we get,
\[
   \Rightarrow \dfrac{{2\pi }}{3} = \dfrac{l}{{2.1}} \\
   \Rightarrow l = \dfrac{{2\pi }}{3} \times \left( {2.1} \right) \\
 \]
On putting the value of \[\pi = \dfrac{{22}}{7}\], we get,
\[
  l = \dfrac{2}{3}\left( {\dfrac{{22}}{7}} \right) \times 2.1 \\
   = 4.4 \\
 \]

Thus, we get the value of the tip covered in 20 minutes is \[4.4\] cm.

Note:
We have used the formula of finding the value of angle when distance and radius are given and compared it with the value of the angle that the tip of minute move in 20 minutes. We have also used that one complete circle is of \[2\pi \]degrees. We must know the conversion of degrees into radians. Do remember the basic formulas like \[\theta = \dfrac{l}{r}\]. As we get the radian for 1 minute, we can easily evaluate for 20 minutes.