Question

Represent the Situation – I with the help of a diagram.

Answer 1

Let the total height of the statue (including base) be $h + 58$ m.
Let point A be at a distance $AB = 80\sqrt{3}$ m from the base of the statue.
Let point C be the top of the statue, and B be the base of the statue.
Then in $\triangle ABC$, $\angle CAB = 60^\circ$, $AB = 80\sqrt{3}$, and $BC = h + 58$. 
We can represent this scenario with a right triangle diagram:

Answer 2

In this case: - The total height of the Statue (including base) is 240 m. - The observer is standing at a height of 40 m above the ground. - The angle of elevation to the top of the Statue from this point is $30^\circ$. Let: - $C$ be the top of the Statue - $B$ be the base of the Statue - $D$ be the observation point 40 m above the ground - $CD = 240 - 40 = 200$ m - $AD$ be the horizontal distance from point D to the Statue This forms a right triangle $\triangle DCA$ with: \[ \angle D = 30^\circ,\quad \text{opposite side } = 200\text{ m} \]

Answer 3

Given:
- Situation–I provides measurements related to the statue and its base.
- We need to calculate:
1. Height of the statue excluding the base.
2. Height of the statue including the base.

Step 1: Understand the problem setup
- Let:
\(h_s\) = height of the statue excluding the base.
\(h_b\) = height of the base.
\(h_t = h_s + h_b\) = total height including the base.

Step 2: Use trigonometric relations from Situation–I
- Typically, Situation–I would provide an angle of elevation \(\theta\) from a certain point at horizontal distance \(d\).
- Using the tangent function:
\[ \tan \theta = \frac{h_t}{d} \] - If the height of the base \(h_b\) is known or calculated separately, then:
\[ h_s = h_t - h_b \]

Step 3: Calculate total height \(h_t\)
- Rearrange tangent formula:
\[ h_t = d \times \tan \theta \] - Substitute the values of \(d\) and \(\theta\) from Situation–I.

Step 4: Calculate height excluding the base \(h_s\)
- If the height of the base \(h_b\) is known from data:
\[ h_s = h_t - h_b \]

Step 5: Interpretation
- Height excluding the base is the part of the statue above the base.
- Height including the base is the total vertical height from the ground.

Note: Use the specific values given in Situation–I to compute numerical answers.

Summary:
\[ h_t = d \times \tan \theta \] \[ h_s = h_t - h_b \] These formulae help find the heights as required.

Answer 4

Given:
- Point \(B\) is located somewhere from which the angle of elevation \(\alpha\) of the top of the base of the statue is observed.
- We need to find:
1. The horizontal distance of point \(B\) from the statue.
2. The value of \(\tan \alpha\).

Step 1: Understand the setup
- The angle \(\alpha\) is the angle of elevation from point \(B\) to the top of the base of the statue.
- The horizontal distance refers to the length of the ground between point \(B\) and the base of the statue.

Step 2: Use trigonometric relations
Let:
- \(h\) = height of the base of the statue.
- \(d\) = horizontal distance of point \(B\) from the statue.
- \(\alpha\) = angle of elevation to the top of the base.

By definition of tangent in a right triangle:
\[ \tan \alpha = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{h}{d} \]

Step 3: Calculate horizontal distance and \(\tan \alpha\)
- If \(h\) and \(\tan \alpha\) are known, then horizontal distance can be found by rearranging:
\[ d = \frac{h}{\tan \alpha} \] - If the horizontal distance \(d\) and height \(h\) are known, then:
\[ \tan \alpha = \frac{h}{d} \]

Step 4: Apply known values from Situation–II (provided by context or figure)
- Substitute the given values for \(h\) and either \(\tan \alpha\) or \(d\) to compute the unknown.

Note: Without the specific values or figure from Situation–II, the problem remains in a general form.

Final Step:
Use the formulae:
\[ \text{Horizontal distance } d = \frac{h}{\tan \alpha} \] and \[ \tan \alpha = \frac{h}{d} \] to calculate the required quantities when values are given.