Question

In 30 minutes, the hours hand of a clock turns through
1.\[\dfrac{\pi }{6}\]radians
2.\[\dfrac{\pi }{{12}}\] radians
3.\[\dfrac{\pi }{{24}}\] radians
4.\[\pi \] radians

Answer 1

Hint: As we know that hours hand complete it’s one rotation in \[{\text{12}}\] hours and in \[{\text{12}}\]hours in turns about \[{{2\pi }}\]angle so rotation per hour can be given as \[\dfrac{{{{2\pi }}}}{{12}}\]radians. As now compared with this proceed for the given question.

Complete step-by-step answer:
As given that in \[30\]minutes we have to calculate hours hand rotation.
As we know that hours hand complete it’s one rotation in \[{\text{12}}\] hours and in \[{\text{12}}\]hours in turns about \[{{2\pi }}\]angle so rotation per hour can be given as \[\dfrac{{{{2\pi }}}}{{12}}\]radians (in \[60\]minutes).
So in \[30\]minutes is,
\[
  {\text{ = (}}\dfrac{{\text{1}}}{{\text{2}}}{\text{)}}\dfrac{{{{2\pi }}}}{{{\text{12}}}} \\
  {\text{ = }}\dfrac{{{\pi }}}{{{\text{12}}}} \\
 \]
Hence, option (B) is our required answer.

Note: Radian describes the plane angle subtended by a circular arc, as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius of the circle.
The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends. One radian is just under \[{\text{57}}{\text{.3}}\]degrees.