Question

The Statue of Unity situated in Gujarat is the world's largest Statue which stands over a 58 m high base. As part of the project, a student constructed an inclinometer and wishes to find the height of Statue of Unity using it. He noted following observations from two places: Situation − I: The angle of elevation of the top of Statue from Place A which is 80`sqrt3` m away from the base of the Statue is found to be 60°. Situation − II: The angle of elevation of the top of Statue from a Place B which is 40 m above the ground is found to be 30° and entire height of the statue, including the base, is found to be 240 m.

Based on the given information, answer the following questions:

1. Represent the Situation – I with the help of a diagram.     [1]
2. Represent the Situation − II with the help of a diagram.    [1]
3. 1. Calculate the height of Statue excluding the base and also find the height including the base with the help of Situation − I.    [2]
                                     OR
   2. Find the horizontal distance of point B (Situation - II) from the Statue and the value of tan α, where α is the angle of elevation of the top of the base of the Statue from point B.    [2]

Answer 1

i. Given Data:

• The base of the Statue is 58 m high.
• The horizontal distance of point A from the base is 80`sqrt3` m.
• The angle of elevation from A to the top of the Statue is 60°.

The situation forms a right-angled triangle with:

• Base = 80`sqrt3` m
• Height = h (excluding the base)
• Hypotenuse connecting A to the top of the Statue.

A diagram should show:

• A vertical line representing the Statue (height = h).
• A horizontal line from A to the Statue’s base.
• A diagonal line from A to the Statue’s top.

ii. Given Data:

• Observer at point B is 40 m above the ground.
• The angle of elevation from B to the top of the Statue is 30°.
• The total height of the Statue including base = 240 m.
• Effective height from B to the top of the Statue = 240 − 40 = 200 m.

A diagram should show:

• A vertical line representing the Statue (height = 240 m).
• A horizontal line from B to the base of the Statue.
• A diagonal line from B to the top of the Statue.

iii. (a) tan 60° = `"opposite"/"adjacent"`

= `h/(80sqrt3)`

Since tan 60° = `sqrt3`, we get:

`sqrt3=h/(80sqrt3)` 

h = `80sqrt3xxsqrt3`

= 80 × 3

= 240 m

Height including the base = 240 m

Height of the Statue excluding the base = 240 − 58 = 182 m.

OR

(b) tan 30° = `"opposite"/"adjacent"`

= `200/x`

Since tan 30° = `1/sqrt3`, we get:

`1/sqrt3=200/x`

x = `200sqrt3`

= 200 × 1.732

= 346.41 m

Thus, horizontal distance of point B from the Statue = 346.41 m.

Finding tan α

• α is the angle of elevation from B to the top of the base.
• The height of the base = 58 m.
• The effective height from B to the top of the base = 58 - 40 = 18 m.

tan α = `18x`

= `18/(200sqrt3)`

= `18/346.41`

= 0.052