A vertical pole of total height 15 m is divided into two parts. The...

Let the distance of the point from the base of the pole be $x$.

Let the angle of elevation of the top of the pole be $\alpha$ and the angle of elevation of the partition be $\beta$.

From the given data, the total height of the pole is $15$ m and the height of the lower part is $5$ m.

Therefore, $\tan \alpha = \dfrac{15}{x}$ and $\tan \beta = \dfrac{5}{x}$.

The angle subtended by the upper part at the point is $\theta = \alpha - \beta$.

We are given that $\tan \theta = \dfrac{1}{2}$.

Using the identity $\tan(\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$, we get:

$\dfrac{1}{2} = \dfrac{\dfrac{15}{x} - \dfrac{5}{x}}{1 + \left(\dfrac{15}{x}\right)\left(\dfrac{5}{x}\right)}$

$\dfrac{1}{2} = \dfrac{\dfrac{10}{x}}{1 + \dfrac{75}{x^2}}$

$\dfrac{1}{2} = \dfrac{10x}{x^2 + 75}$

$x^2 + 75 = 20x$

$x^2 - 20x + 75 = 0$

$(x - 5)(x - 15) = 0$

Thus, $x = 5$ or $x = 15$.

Answer: $5$ m and $15$ m