Question

The minute hand of a clock is 10 cm long. Find the area of the face of the clock described by the minute hand between 9 A.M. and 9:35 A.M.
A. $ 90.165\text{ c}{{\text{m}}^{2}} $
B. $ 112.6\text{ c}{{\text{m}}^{2}} $
C. $ 156.4\text{ c}{{\text{m}}^{2}} $
D. $ 172.78\text{ c}{{\text{m}}^{2}} $

Answer 1

Hint: We first find the relation between the minute hand and the circular area created by its circulation. We try to find the time required for the minute hand to complete a full rotation and the area it creates for the full rotation. The ratio of area to time is linear. Keeping that in mind we find the area covered by the minute hand for 35 minutes.

Complete step by step answer:
The minute hand of a clock is 10 cm long. The area created by the motion of the minute hand is a circular area as the minute hand works as a radius and it’s fixed at a point.
The full area is created only when the minute hand covers the full area one time. One-time circulation of the minute hand is considered as the completion of 1 hour.
In 1 hour, the minute hand covers the surface area of a circle with a radius of 10 cm.
We know that area of a circle with radius r cm is $ \pi {{r}^{2}}\text{ c}{{\text{m}}^{2}} $ .
We put the value of $ r=10 $ and get the circular area as $ \pi {{\left( 10 \right)}^{2}}=100\pi $ .
Now we need to find the circular area covered by the minute hand when it goes from 9 A.M. and 9:35 A.M.
So, instead of 60 minutes the hand only covers 35 minutes. The area changes with the same ratio.
60 minutes rotation for the minute hand is equal to $ 100\pi $ are of the circle and we need to find the area for 35 minutes.
The area will be \[\dfrac{100\pi }{60}\times 35=\dfrac{165\pi }{3}=\text{172}\text{.78 c}{{\text{m}}^{2}}\]. The correct option is D.

Note:
 We could also have used angular rotation instead of minutes. The total angular rotation for the minute hand in the whole 60 minutes will be $ {{360}^{\circ }} $ . So, for every minute we have $ {{6}^{\circ }} $ angular rotation. Then we find it for 35 minutes. As the ratio is still linear, we find the area according to that.