Given:
\[ V = \frac{2\pi R}{T} \]
The maximum height attained by the particle is given by:
\[ H = \frac{v^2 \sin^2 \theta}{2g} \]
We are given that:
\[ 4R = \frac{4\pi^2 R^2 \sin^2 \theta}{T^2 \cdot 2g} \]
Simplifying:
\[ \sin^2 \theta = \frac{2gT^2}{\pi^2 R} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{2gT^2}{\pi^2 R}} \]
Thus:
\[ \theta = \sin^{-1} \left( \sqrt{\frac{2gT^2}{\pi^2 R}} \right) \]
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: