Question

The length of a minute hand of a wall clock is $9$ cm. What is the area swept (in sq. cm) by the minute hand in $20$ minutes? (Take $\text{ }\!\!\pi\!\!\text{ }$= 3.14)
(A) $88.78$
(B) $84.78$
(C) $67.74$
(D)$57.78$

Answer 1

Hint: To solve this question we need to have a concept of angle. To solve this kind of question the first step is to find the angle covered by the minute hand in $20$ minutes. To solve the question we must know the formula for area of the sector which is $\dfrac{\theta }{2}{{r}^{2}}$, where $\theta $ is the angle of the sector and r is the radius of the sector.

Complete step by step answer:
The question asks us to find the area which is swept by the minute hand in $20$ minutes if the length of the minute hand of a clock wall is given as $9$ cm. Now we know that a minute hand moves around the whole clock in 60 minutes which means it covers ${{360}^{\circ }}$ which is $2\pi $ in radian.
We know that the area of the circle is $\pi {{r}^{2}}$ . Here $r$ is the radius of the circle. While the area of a sector is $\dfrac{\theta }{2}{{r}^{2}}$ , where $\theta $ is the angle inscribed in the sector.
Now in this question the minute hand covers $20$ so the angle thus covered by the minute hand in $20$ minutes is
$\Rightarrow \dfrac{2\pi }{60}\times 20$
$= \dfrac{2\pi }{3}$
So the area swept by the minute hand in $20$ minutes is
\[= \dfrac{\left( \dfrac{2\pi }{3} \right)}{2}{{r}^{2}}\]
$= \dfrac{\pi }{3}{{r}^{2}}$
Here $r$ is the radius which is the length of the minute hand given in this problem which is $9$ cm. Putting the values in the above formula we get:
$= \dfrac{\pi }{3}{{9}^{2}}$
We need to take $\pi $ as $3.14$ as given in the question
$= \dfrac{3.14}{3}\times 9\times 9$
$= 3.14\times 9\times 3$
On calculating it further we get:
$= $ $84.78$
$\therefore $ The length of a minute hand of a wall clock is $9$ cm. The area swept (in sq. cm) by the minute hand in $20$ minutes is $\left( B \right)84.78$ .

So, the correct answer is “Option B”.

Note: Find the angles always in radian form as it becomes easier to find the area of the sector. Do remember that the angle ${{360}^{\circ }}=2\pi $. These small things help a lot while solving the problem. To complete a rotation a minute hand takes 60 minutes which inscribes an angle $2\pi $.