Question

The minute hand of a clock is 14cm long. If it moves between 8:00 AM and 8:45 AM. What is the area covered by it on the face of the clock?
A. $512c{m^2}$
B. $462c{m^2}$
C. $264c{m^2}$
D. $196c{m^2}$

Answer 1

Hint: We will take the length of the hand of the clock as radius of the circle. Since it takes 60 minutes to cover ${360^ \circ }$, find the angle covered from 8:00 AM and 8:45 AM, that is in 45 minutes. Then, apply the formula of area of sector, $\dfrac{\theta }{{360}}\pi {r^2}$, where $r$ is the radius of the sector and $\theta $ is the angle to determine the required area.

Complete step-by-step answer:
We are given that the hand of the clock is 14cm.
We know that the hand in a clock moves in circular motion.
Then, the radius of the circle formed is 14cm.
Also, total angle covered in 1 revolution or in 1 hour is ${360^ \circ }$
And there are 60 minutes in 1 revolution.
That is it takes 60 minutes to cover ${360^ \circ }$
We can calculate the angle formed in 1 minute by dividing ${360^ \circ }$ by 60
$\dfrac{{{{360}^ \circ }}}{{60}} = {6^ \circ }$
The hand of the clock has moved from 8:00 AM to 8:45 AM.
Hence, the clock has covered 45 minutes
Thus, the angle formed by the hand in 45 minutes is ${6^ \circ } \times 45 = {270^ \circ }$
It is known that the area of the sector is given by $\dfrac{\theta }{{360}}\pi {r^2}$, where $r$ is the radius of the sector and $\theta $ is the angle.
We will calculate the area of the sector when radius is 14cm and angle is ${270^ \circ }$
$\dfrac{{270}}{{360}}\pi {\left( {14} \right)^2}$
On substituting the value of $\pi = \dfrac{{22}}{7}$, we will get,
$\dfrac{{270}}{{360}}\left( {\dfrac{{22}}{7}} \right){\left( {14} \right)^2} = 462c{m^2}$
Thus, area covered by it on the face of the clock between 8:00 AM and 8:45 AM is $462c{m^2}$
Hence, option B is correct.

Note: We can alternatively find the area of the whole circle formed and then subtract the area of the sector from 8:45 AM to 9:00 AM. And the sector is a part of a circle which is enclosed by an arc and its two radii.