Question

The angle of elevation of the top of a tower from the top of 5m tall building is 60 and angle of depression of the bottom of a tower is 45 . Find the height of the tower and distance between the tower and building .
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Answer 1

Answer:

Height of the tower [EC] = 13.66 metres.

Distance between the Tower and the Building [BC] = 5 metres.

Pre-requisite knowledge:

• Angle of elevation refers to the angle made between the horizontal line and the line of sight, where the object being viewed is located above the eye level.

• Angle of depression refers to the angle made between the horizontal line and the line of sight, where the object being viewed is located below the eye level.

• Trigonometric ratios and values of certain angles.

Step-by-step explanation:

➝ Height of the building = AB = 5 metre.

➝ Height of the tower = EC

➝ ∠ of Elevation from the top of AB to the top of EC = ∠EAD = 60°

➝ ∠ of Depression from the top of AB to the base of EC = ∠DAC = 45°

➝ ∠DAC = ∠ACB = 45° [Alternate interior angles are equal]

Calculating the distance between the Tower & the Building:

In ΔABC,

• BC = Distance between the tower and the building.

• AB = 5m

• ∠ACB = 45°

With the help of trigonometric ratios we can say that;

$\sf \Longrightarrow tan\theta = \dfrac{Side \ opposite \ to \ \theta}{Side \ adjacent \ to \ \theta}$

Here, θ = 45°.

$\sf \Longrightarrow tan45^{\circ} = \dfrac{AB}{BC}$

$\sf \Longrightarrow tan45^{\circ} = \dfrac{5}{BC}$

We know that tan45° = 1. Therefore,

$\sf \Longrightarrow 1 = \dfrac{5}{BC}$

$\sf \Longrightarrow \textsf{\textbf{BC = 5m}}$

Therefore, the distance between the tower and building is 5 metres.

Calculating the height of the tower:

In ΔAED,

• ∠EAD = 60°

• BC = AD = 5m

With the help of trigonometric ratios we can say that;

$\sf \Longrightarrow tan\theta = \dfrac{Side \ opposite \ to \ \theta}{Side \ adjacent \ to \ \theta}$

Here, θ = 60°.

$\sf \Longrightarrow tan60^{\circ} = \dfrac{ED}{AD}$

$\sf \Longrightarrow tan60^{\circ} = \dfrac{ED}{5}$

We know that tan60° = √3. Therefore,

$\sf \Longrightarrow \sqrt{3} = \dfrac{ED}{5}$

$\sf \Longrightarrow ED = 5\sqrt{3}$

$\sf \Longrightarrow ED = 5 \times1.732$

$\sf \Longrightarrow \textsf{\textbf{ED = 8.66m}}$

We know that;

➝ Height of the tower = ED + DC

➝ Height of the tower = 8.66 + DC

[DC = AB = 5m]

➝ Height of the tower = 8.66 + 5

➝ Height of the tower = 13.66m

Therefore, the height of the tower is 13.66 metres.