Question

The angles of depression of the top and the foot of a 9 m tall building from the top of a multi-storeyed building are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
(Use \(\sqrt{3} = 1.73\))

Hint:

Use the angle of depression concept and \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) in both cases to set up two equations and solve them simultaneously.

Answer 1

Given:
Height of smaller building = 9 m
Angle of depression to top of smaller building = \(30^\circ\)
Angle of depression to foot of smaller building = \(60^\circ\)
Let height of multi-storeyed building = \(h\) m
Distance between the two buildings = \(d\) m

Step 1: Draw a diagram and identify right triangles
From the top of the taller building, draw horizontal line of sight.
Let the top and foot of smaller building be points A and B respectively.
Let the top of taller building be point C.
The angles of depression are given:
\(\angle CAD = 30^\circ\) (to top A)
\(\angle CBD = 60^\circ\) (to foot B)

Step 2: Use right triangle properties
Since angles of depression equal angles of elevation from the foot of the smaller building:
In \(\triangle ADC\) (right triangle):
\[ \tan 30^\circ = \frac{\text{height difference}}{\text{distance}} = \frac{h - 9}{d} \]
In \(\triangle BDC\) (right triangle):
\[ \tan 60^\circ = \frac{h}{d} \]

Step 3: Write equations using \(\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{1}{1.73}\) and \(\tan 60^\circ = \sqrt{3} = 1.73\)
\[ \tan 30^\circ = \frac{h - 9}{d} = \frac{1}{1.73} \Rightarrow h - 9 = \frac{d}{1.73} \quad ...(1) \]
\[ \tan 60^\circ = \frac{h}{d} = 1.73 \Rightarrow h = 1.73 d \quad ...(2) \]

Step 4: Substitute \(h\) from (2) into (1)
\[ 1.73 d - 9 = \frac{d}{1.73} \]
Multiply both sides by 1.73 to clear denominator:
\[ 1.73 \times 1.73 d - 9 \times 1.73 = d \Rightarrow 3 d - 15.57 = d \]
\[ 3 d - d = 15.57 \Rightarrow 2 d = 15.57 \Rightarrow d = \frac{15.57}{2} = 7.785 \text{ m} \]

Step 5: Calculate height \(h\) using (2)
\[ h = 1.73 \times 7.785 = 13.46 \text{ m} \]

Final Answer:
Height of multi-storeyed building = 13.46 m
Distance between the two buildings = 7.785 m