Question

The length of a minute hand of a clock is 14 cm. If the minute hand moves from 1 to 10 on the dial, then find the area swept by the minute hand.
A. 462
B. 154
C. 308
D. 616

Answer 1

Hint: We will first find the angle it covers in moving from 1 to 10. Now, we will use the formula \[A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\], where A is the area required, r is the radius that is the length of minute hand and $\theta $ is the angle covered in degrees. Thus, we will have the area required.

Complete step-by-step answer:
We are given that the minute hand of a clock is 14 cm long.
We need to find the area swept by it in moving from 1 to 10.
We know that Area is given by \[A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\], where A is the area required, r is the radius that is the length of minute hand and $\theta $ is the angle covered in degrees.
We also know that 1 hr = 60 minutes.
Hence, a minute hand will cover the whole \[{360^ \circ }\] in 60 minutes.
So, 60 minutes are equivalent to \[{360^ \circ }\].
$ \Rightarrow $ 1 minute is equivalent to $\dfrac{{{{360}^ \circ }}}{{60}} = {6^ \circ }$.
Now, when the minute hand is on 1, the minute count is 5 minutes and when it is on 10, the minute count is 50 minutes. Hence, the minutes it covered while moving from 1 to 10 is 50 – 5 minutes = 45 minutes.
Since, we just found out: $ \Rightarrow $ 1 minute is equivalent to ${6^ \circ }$.
Therefore, 45 minutes is equivalent to $45 \times {6^ \circ } = {270^ \circ }$. ………(1)
Now, coming to the formula \[A = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\].
\[ \Rightarrow A = \dfrac{{{{270}^ \circ }}}{{{{360}^ \circ }}} \times \pi {r^2}\]
Taking 90 common from numerator and denominator on RHS:-
\[ \Rightarrow A = \dfrac{3}{4} \times \pi \times {14^2}\]
Simplifying it:-
\[ \Rightarrow A = \dfrac{3}{4} \times \pi \times 196 = 3 \times \pi \times 49\]
Putting the value of $\pi = \dfrac{{22}}{7}$, we will get:-
\[ \Rightarrow A = 3 \times \dfrac{{22}}{7} \times 49 = 462c{m^2}\].
Hence, the answer is 462 square cm.

So, the correct answer is “Option A”.

Note: Students must note that they may use any value of $\pi $ among $\dfrac{{22}}{7}$ or 3.14 if not mentioned in the question. If given specifically, you must use that value only, otherwise the answer may vary.
The students must always remember to change the angles from degree to change the angle in $\pi $. They make mistakes by forgetting to do so. If you want to use the value in degree, you must divide by 360 in the formula of area itself. That will reduce 1-2 steps.