Question

The minute hand of a clock is \[12cm\]long. Then the area of the face of the clock described by the minute hand in 35 minutes is
A.$210\;c{m^2}$
B.$264\;c{m^2}$
C.$144\;c{m^2}$
D.$200\;c{m^2}$

Answer 1

Hint: We use the formulas related to clock type problems which we need to know to solve such types of problems. We need to know that a minute hand completes the whole cycle in 60 min and also minute hand moves by ${6^ \circ }$ in a minute. Also area of arc which makes an angle ${\theta ^ \circ }$ at the center is given as-
$A = \dfrac{\theta }{{{{360}^ \circ }}} \times \left( {\pi {r^2}} \right)$


Complete step by step solution:

The given minute hand is $12cm$ long

The minute hand covers a whole cycle in 60 min, then in 35 min the angle covered is

$\begin{gathered}
  60\;{\text{mins}} \to {\text{36}}{{\text{0}}^ \circ } \\
  1\;{\text{min}} \to {6^ \circ } \\
  35\;\min \to 35 \times {6^ \circ } = {210^ \circ } \\
\end{gathered} $

The radius is $r = 12\;cm$

So, the area covered in 35 min is
$A = \dfrac{\theta }{{{{360}^ \circ }}} \times \left( {\pi {r^2}} \right)$
Substituting the values we get
$\begin{gathered}
   = \dfrac{{{{210}^ \circ }}}{{{{360}^ \circ }}} \times \left( {\pi {{\left( {12} \right)}^2}} \right) \\
   = \dfrac{7}{{12}} \times \left( {\dfrac{{22}}{7} \times 12 \times 12} \right) = 22 \times 12 \\
  A = 264\;c{m^2} \\
\end{gathered} $
Thus the area covered is$A = 264\;c{m^2}$.

Hence the correct option is B.


Note: The area of a section of circle for any given angle is given by above formula which values should be substituted very carefully. We should have basic formulas and tips for clocks to solve such types of problems. Calculation mistakes should be avoided.