Question

Totem poles are made from large trees. These poles are carved with symbols or figures and are mostly found in Western Canada and northwestern United States. Two such poles of equal heights are standing 28 m apart. From a point somewhere between them in the same line, the angles of elevation of the top of the two poles are 60° and 30° respectively.

Based on the above information, answer the following questions.

1. Draw a neat well-labelled figure based on the above information.
2. Find the height of the poles.
3. If the distances of the top of the poles from the point of observation are taken as p and q then find a relation between p and q.
4. Find the location of the point of observation.

Answer 1

i. Figure:

Let the two poles be (AB) and (CD) of equal height (h) metres, standing 28 m apart on the ground BD. Let (E) be the point of observation between them such that \(\angle AEB = 60^\circ\) and \(\angle CED = 30^\circ\).

ii. Height of the poles

Let \(BE = x\) m, then \(ED = (28 - x)\) m.

In right \(\triangle ABE\): \[\tan 60^\circ = \frac{AB}{BE}\]

\[\Rightarrow \sqrt{3} = \frac{h}{x}\]

\[\Rightarrow h = \sqrt{3}x \quad \ldots(1)\]

In right \(\triangle CDE\): \[\tan 30^\circ = \frac{CD}{DE}\]

\[\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{28 - x}\]

\[\Rightarrow h = \frac{28 - x}{\sqrt{3}} \quad \ldots(2)\]

Equating (1) and (2):

\[\sqrt{3}x = \frac{28 - x}{\sqrt{3}}\] 

\[\Rightarrow 3x = 28 - x\]

\[\Rightarrow 4x = 28\]

\[\Rightarrow x = 7\]

Substituting \(x = 7\) in (1): \[h = 7\sqrt{3} \text{ m}\]

Height of each pole = \(7\sqrt{3}\) m.

iii. Relation between (p) and (q)

Let \(AE = p\) and \(CE = q\).

In \(\triangle ABE\): \[\sin 60^\circ = \frac{AB}{AE}\]

\[\Rightarrow \frac{\sqrt{3}}{2} = \frac{h}{p}\]

\[\Rightarrow h = \frac{\sqrt{3}p}{2}\]

In \(\triangle CDE\): \[\sin 30^\circ = \frac{CD}{CE}\]

\[\Rightarrow \frac{1}{2} = \frac{h}{q}\]

\[\Rightarrow h = \frac{q}{2}\]

Equating the two expressions for (h):

\[\frac{\sqrt{3}p}{2} = \frac{q}{2}\]

\[\Rightarrow q = \sqrt{3}p\]

iv. Location of the point of observation

From (ii), \(BE = x = 7\) m. Hence, the point of observation (E) is 7 m from the first pole (the one with 60° elevation).