Hint: We are asked to find the height of the first pole. As we have that top of the second pole, make an angle of the depression on the top of the first pole. So, clearly, we get the second pole as taller. We will first draw our structure and apply the given details as the height of the second pole is 24m, the base distance is 15m. Then using trigonometry ratio \[\tan \theta \] we will find the value of the height of the pole first.

Complete step-by-step answer:

We are given that there are two poles which are at a distance of 15m horizontally. The angle of depression of the top at the first pole as seen from the second pole’s top is \[{{30}^{\circ }}.\] So, we can clearly understand from here that the second pole is larger than the first pole. Also, we have the height of the second pole as 24m. Using these, we can construct the structure.

AC = height of the second pole = 24m

Let us consider the height of the first pole as hm. So,

\[Eh=hm\]

Now, AC = AB + BC

As AC = 24m and BC = Eh = h. So, we get,

\[\Rightarrow 24=AB+h\]

\[\Rightarrow AB=24-h\]

Also, as we have the horizontal distance as 15, we will get,

\[CD=15m\]

So, therefore,

\[BE=CD\]

\[\Rightarrow BE=15m\]

Now, in the triangle ABE, we use the trigonometry ratio to find the value of h.

In triangle ABE,

\[\dfrac{AB}{BE}=\tan {{30}^{\circ }}\]

Putting the values of AB = 24 – h and BE = 15, we will get,

\[\Rightarrow \dfrac{24-h}{15}=\dfrac{1}{\sqrt{3}}\]

\[\Rightarrow 24-h=\dfrac{15}{\sqrt{3}}\]

Rationalizing the denominator, we get,

\[\Rightarrow 24-h=\dfrac{15\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}\]

\[\Rightarrow 24-h=\dfrac{15\sqrt{3}}{3}\]

On further simplification, we will get,

\[\Rightarrow 24-h=5\sqrt{3}\]

\[\Rightarrow h=24-5\sqrt{3}\]

We know that, \[\sqrt{3}=1.732\]

So, we get,

\[h=24-5\times 1.732\]

\[\Rightarrow h=15.34m\]

So, we get that the height of the first pole was 15.34m.

Note: The key point to remember here is that first, we have to understand which pole is bigger as if we have considered second as the smaller pole, so our structure would be like

So, to find the height of the pole one, we will first find h and then add it to 24m which will lead us to the wrong solution.