A player sitting on the top of a tower of height 20 m observes the angle of depression of a ball lying on the ground as'60°. Find the distance between the foot of the tower and the balL Take `sqrt(3) = 1.732`

Answer

To solve the problem step by step, we will use trigonometric ratios and the information given in the question.

### Step 1: Understand the problem
We have a tower of height 20 m, and the angle of depression from the top of the tower to the ball on the ground is 60°. We need to find the horizontal distance (let's call it BC) between the foot of the tower and the ball.

### Step 2: Draw a diagram
Draw a right triangle where:
- Point A is the top of the tower.
- Point B is the foot of the tower.
- Point C is the position of the ball on the ground.
- The height of the tower (AB) is 20 m.
- The angle of depression (∠ACB) is 60°.

Since the angle of depression from A to C is 60°, the angle ∠CAB (the angle of elevation from C to A) is also 60°.

### Step 3: Use the tangent function
In the right triangle ABC:
- The opposite side (height of the tower) is AB = 20 m.
- The adjacent side (distance from the foot of the tower to the ball) is BC.

Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here, \(\theta = 60°\), so:
\[ \tan(60°) = \frac{AB}{BC} \]

### Step 4: Substitute known values
We know that:
- \(\tan(60°) = \sqrt{3}\)
- \(AB = 20\) m

Substituting these values into the equation:
\[ \sqrt{3} = \frac{20}{BC} \]

### Step 5: Solve for BC
Rearranging the equation to find BC:
\[ BC = \frac{20}{\sqrt{3}} \]

### Step 6: Substitute the value of \(\sqrt{3}\)
Given that \(\sqrt{3} \approx 1.732\), we substitute this value:
\[ BC = \frac{20}{1.732} \]

### Step 7: Calculate the value
Now, perform the division:
\[ BC \approx 11.54 \text{ m} \]

### Conclusion
The distance between the foot of the tower and the ball is approximately **11.54 meters**.

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