Two people standing on the same side of a tower in a straight line with it, measure the angles of elevation of the top of the tower as `25^(@) and 50^(@)` respectively. If the height of the tower is 70 m, find the distance between the two people

Answer

To solve the problem step by step, we will use trigonometric ratios to find the distances from each person to the base of the tower and then calculate the distance between the two people.

### Step 1: Understand the Problem
We have a tower of height 70 m, and two people (A and B) are standing on the same side of the tower. The angles of elevation from person A and person B to the top of the tower are 25° and 50°, respectively. We need to find the distance between the two people.

### Step 2: Draw the Diagram
1. Draw a vertical line representing the tower PQ, where PQ = 70 m.
2. Mark point A and point B on the ground, both on the same side of the tower.
3. Draw lines from A and B to the top of the tower P, forming angles of elevation of 25° and 50° respectively.

### Step 3: Define Variables
- Let AQ = x (the distance from person A to the base of the tower).
- Let BQ = y (the distance from person B to the base of the tower).

### Step 4: Use the Tangent Function for Person B
For person B, we can use the tangent function:
\[ \tan(50°) = \frac{PQ}{BQ} = \frac{70}{y} \]
Rearranging gives:
\[ y = \frac{70}{\tan(50°)} \]
Calculating \(\tan(50°)\):
\[ \tan(50°) \approx 1.1917 \]
Thus:
\[ y = \frac{70}{1.1917} \approx 58.73 \text{ m} \]

### Step 5: Use the Tangent Function for Person A
For person A, we can use the tangent function:
\[ \tan(25°) = \frac{PQ}{AQ} = \frac{70}{x} \]
Rearranging gives:
\[ x = \frac{70}{\tan(25°)} \]
Calculating \(\tan(25°)\):
\[ \tan(25°) \approx 0.4663 \]
Thus:
\[ x = \frac{70}{0.4663} \approx 150.11 \text{ m} \]

### Step 6: Calculate the Distance Between the Two People
The distance between the two people (AB) is given by:
\[ AB = AQ - BQ = x - y \]
Substituting the values we found:
\[ AB = 150.11 - 58.73 \approx 91.38 \text{ m} \]

### Final Answer
The distance between the two people is approximately **91.38 m**.

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