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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 $9$ am and $9:35$ am.

$\begin{align}&\color{green}{a.)183.3\quad cm^2}\\ &b.)366.6\quad cm^2\\ &c.)244.4\quad cm^2\\ &d.)188.39\quad cm^2\\ \end{align}$

I tried

$Area=\dfrac{30}{360} \times \pi\times 10^2=26.17\quad cm^2$

what is the mistake ?

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  • $\begingroup$ What's the angle? Definitely not $30^\circ$... $\endgroup$ – b00n heT May 17 '15 at 21:04
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What makes you think it is $\frac{30}{360}$ ?

The correct rate is $\frac{35}{60}$, since 35 minutes pass and one full circle is 60 minutes.

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  • $\begingroup$ Oh ok i see it now. thnx $\endgroup$ – R K May 17 '15 at 21:06
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When the minute hand moved from 9 am to 9:35 am,

It moved $\frac7{12}$ cicrle,that is,

it moved $\frac{7}{12}\cdot360=210$ degree NOT $30$ degree.

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In a 12 hour clock, we know that $$1 \space minute=\frac{360}{60}=6^o=\frac{6\times \pi}{180}=\frac{\pi}{30} \space radian$$

Hence, the area swept by the minute hand between 9 am to 9:35 am is $$=\frac{1}{2}\times(\text{total angle swept})\times(\text{length of minute hand})^2=\frac{1}{2}\times\left(35\times \frac{\pi}{30}\right)\times(10)^2=\frac{175\pi}{3}\approx183.2595715 \space cm^2$$

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The fraction of full angle in minutes time =$ \dfrac{35}{60} $

The fraction of swept area =$ \dfrac{35}{60} \pi 10^{2} \approx 183.26. $

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