A magnetic compass needle oscillates 30 times per minute at a place...

Let ${f}_{1}$ and ${f}_{2}$ be the respective oscillation frequencies respectively for $45 °$ and $30 °$ dip. Then,

${f}_{1} = \dfrac{1}{2\pi}\sqrt{\dfrac{M{B}_{1}\cos 45 °}{I}}$

${f}_{2} = \dfrac{1}{2\pi}\sqrt{\dfrac{M{B}_{2}\cos 30 °}{I}}$

$\dfrac{{f}_{1}}{{f}_{2}} = \sqrt{\dfrac{{B}_{1}\cos 45 °}{{B}_{2}\cos 30 °}}$

$⇒ \dfrac{{30}^{2}}{{40}^{2}} = \dfrac{{B}_{1}\cos 45 °}{{B}_{2}\cos 30 °}$

$⇒ \dfrac{{B}_{1}}{{B}_{2}} = \dfrac{900 \times \cos 30 °}{1600 \times \cos 45 °}$

$⇒ \dfrac{{B}_{1}}{{B}_{2}} = 0.688 \approx 0.7$