A vertical tower is 20 m high. A man standing at some distance from the tower knows that the cosine of the angle of elevation of the top of the tower is 0.53. How far is he standing from the foot of the tower ?

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Step 1: Understand the Problem
We have a vertical tower of height 20 m and a man standing at a distance from the tower. The cosine of the angle of elevation of the top of the tower from the man's position is given as 0.53. We need to find the distance of the man from the foot of the tower.

### Step 2: Draw the Diagram
Let's visualize the situation:
- Let the point at the top of the tower be A.
- Let the foot of the tower be B.
- Let the position of the man be C.
- The height of the tower (AB) is 20 m.
- The distance from the foot of the tower to the man (BC) is x m.

### Step 3: Set Up the Relationship
From the right triangle ABC, we know:
- AB = 20 m (height of the tower)
- BC = x m (distance from the tower)
- AC = hypotenuse

Using the cosine of the angle of elevation (θ):
\[ \cos(θ) = \frac{BC}{AC} \]
Given that \(\cos(θ) = 0.53\), we can write:
\[ 0.53 = \frac{x}{AC} \]

### Step 4: Use Pythagorean Theorem
To find AC, we can use the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{20^2 + x^2} = \sqrt{400 + x^2} \]

### Step 5: Substitute AC in the Cosine Equation
Substituting AC in the cosine equation:
\[ 0.53 = \frac{x}{\sqrt{400 + x^2}} \]

### Step 6: Square Both Sides
To eliminate the square root, we square both sides:
\[ 0.53^2 = \frac{x^2}{400 + x^2} \]
Calculating \(0.53^2\):
\[ 0.2809 = \frac{x^2}{400 + x^2} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 0.2809(400 + x^2) = x^2 \]
Expanding this:
\[ 112.36 + 0.2809x^2 = x^2 \]

### Step 8: Rearrange the Equation
Rearranging the equation:
\[ x^2 - 0.2809x^2 = 112.36 \]
\[ 0.7191x^2 = 112.36 \]

### Step 9: Solve for x^2
Dividing both sides by 0.7191:
\[ x^2 = \frac{112.36}{0.7191} \approx 156.25 \]

### Step 10: Find x
Taking the square root of both sides:
\[ x = \sqrt{156.25} = 12.5 \text{ m} \]

### Conclusion
The distance of the man from the foot of the tower is **12.5 meters**.

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