Question

Easy Questions!!!

Evaluate the approximate area covered by hour hand in 1 hour, where the length of hour hand of a clock is 7 cm.

(a) 9 cm²
(b) 11 cm²
(C) 13 cm²
(d) 15 cm²

Rohit and Mohit run 5 km race on a 400 m athletics track. Rohit runs the distance in 25 minutes while Mohit takes 20 minutes. If they start from the same point simultaneously and run in the same sense, how many times will Mohit overtake Rohit during the race?

a) 1
b) 2
c) 3
d) 4​​

Answer 1

$\large\underline{\sf{Solution-1}}$

Evaluate the approximate area covered by hour hand in 1 hour, where the length of hour hand of a clock is 7 cm.

(a) 9 cm²

(b) 11 cm²

(C) 13 cm²

(d) 15 cm²

CALCULATIONS

Given that,

Hour hand of a clock = 7 cm

We know,

In 12 hours, angle subtended at the centre is 360°

So, In 1 hour, angle subtended at the centre is 30°.

So, required area covered by hour hand is equals to area of sector of radius 7 cm having sector angle 30°.

We know,

$\rm :\longmapsto\:\boxed{\tt{ \: \: Area_{(sector)} \: = \: \frac{\pi \: {r}^{2} \theta \: }{360} \: }}$

So, on substituting the values, we get

$\rm :\longmapsto\:Area_{(sector)} = \dfrac{22}{7} \times 7 \times 7 \times \dfrac{30}{360}$

$\rm :\longmapsto\:Area_{(sector)} = 11 \times 7 \times \dfrac{1}{6}$

$\rm :\longmapsto\:Area_{(sector)} = \dfrac{77}{6}$

$\bf\implies \:Area_{(sector)} = 13 \: {cm}^{2} \: (approx.) \:$

$\green{\rm\implies \:\boxed{\tt{ \: \: Option \: (c) \: is \: correct \: \: }}}$

$\red{\large\underline{\sf{Solution-2}}}$

Rohit and Mohit run 5 km race on a 400 m athletics track. Rohit runs the distance in 25 minutes while Mohit takes 20 minutes. If they start from the same point simultaneously and run in the same sense, how many times will Mohit overtake Rohit during the race?

a) 1

b) 2

c) 3

d) 4

CALCULATIONS

Total distance to be covered = 5 km = 5000 m

One round is of 400 m.

Total number of rounds = 5000 ÷ 400 = 12.5 rounds.

Rohit cover 400 m in 25 minutes

Mohit cover 400 m in 20 minutes.

So, it means

Time taken by Rohit to cover 12.5 rounds = 25 × 12.5 = 312.5 minutes.

And,

Time taken by Mohit to cover 12.5 rounds = 20 × 12.5 = 250 minutes.

Now,

Rohit cover 400 m in 25 minutes

Mohit cover 400 m in 20 minutes.

So, it means they meet together first time after they start = LCM ( 20 and 25 ) = 100 minutes.

So, it means again after another 100 minutes.

After this, Mohit finish in 50 minutes. So, they can't meet again.

So, it means 2 times, Mohit overtake the Rohit.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

$\begin{gathered}\begin{gathered}\boxed{\begin {array}{cc}\\ \dag\quad \Large\underline{\bf Formulas\:of\:Areas:-}\\ \\ \star\sf Square=(side)^2\\ \\ \star\sf Rectangle=Length\times Breadth \\\\ \star\sf Triangle=\dfrac{1}{2}\times Breadth\times Height \\\\ \star \sf Scalene\triangle=\sqrt {s (s-a)(s-b)(s-c)}\\ \\ \star \sf Rhombus =\dfrac {1}{2}\times d_1\times d_2 \\\\ \star\sf Rhombus =\:\dfrac {1}{2}d\sqrt {4a^2-d^2}\\ \\ \star\sf Parallelogram =Breadth\times Height\\\\ \star\sf Trapezium =\dfrac {1}{2}(a+b)\times Height \\ \\ \star\sf Equilateral\:Triangle=\dfrac {\sqrt{3}}{4}(side)^2\end {array}}\end{gathered}\end{gathered}$

Answer 2

Given: the length of hour hand of a clock is 7 cm.

To find: the approximate area covered by hour hand in 1 hour,

Solution 1:

Know that, In 12 hours, angle subtended at the centre is $360^\circ$ . So, In 1 hour, angle subtended at the centre is $30^\circ$  then, the required area covered by hour hand is equals to area of sector of radius 7 cm having sector angle $30^\circ$.

Find the area covered by the hour hand in 1 hour.

$\[A = \frac{\theta }{{{{360}^ \circ }}} \times \pi {r^2}\]$

$\[ \Rightarrow A = \frac{{60}}{{360}} \times \frac{{22}}{7} \times 7 \times 7\]$

$\[ \Rightarrow A = 11 \times 7 \times \frac{1}{6}\]$

$\[\begin{array}{l} \Rightarrow A = \frac{{77}}{6}\\ \Rightarrow A = 12.8 \approx 13c{m^2}\end{array}\]$

Therefore, the area covered by the hour hand in 1 hour is $13cm^2$ approximately.

Hence, the correct answer is option (C). i.e., $13cm^2$ .

-------------------------------------------------------------------------------------------------------------Given:

Rohit and Mohit run 5 km race on a 400 m athletics track.

Time taken by Rohit $=25 minutes$

Time taken by Mohit$=20 minutes$

To find: how many times will Mohit overtake Rohit during the race?

Solution 2:

Note that from the question,

Total distance to be covered $= 5 km$

                                                $= 5000 m$            

One round of the track is of $400 m.$

Total number of rounds $\[ = \frac{{5000}}{{400}}\]$

                                       $\[ = 12.5rounds\]$

Know that from the question, Rohit cover 400 m in 25 minutes  and Mohit cover 400 m in 20 minutes.

Find the time taken by Rohit to cover $12.5$rounds.

Time taken by Rohit to cover 12.5 rounds$= 25 \times12.5$

                                                                     $= 312.5 minutes.$

Find the time taken by Mohit to cover $12.5$rounds.

Time taken by Mohit to cover 12.5 rounds $= 20 \times12.5$

                                                                      $= 250 minutes.$

Understand that, they meet together first time after they start in,

$= LCM ( 20 and 25 ) \\= 100 minutes.$

Therefore, they will meet again after another 100 minutes. and After this, Mohit finish in 50 minutes. Thus, they can't meet again.

Hence, Mohit overtake Rohit only 2 times.

Hence, the correct answer is option (b). i.e., 2.