P and Q are two points on the opposite sides of a 90 m high tower AB .The base B, of the tower AB , and points P and Q as observed from top A of tower AB are ` 60 ^(@) and 30 ^(@) ` respectively. Find correct to the nearest .the distance between P and Q.

Answer

To find the distance between points P and Q, we can follow these steps:

### Step 1: Understand the problem and draw a diagram
We have a tower AB with height 90 m. Points P and Q are on opposite sides of the tower. The angle of depression from point A (top of the tower) to point P is 60 degrees, and to point Q is 30 degrees.

### Step 2: Label the diagram
Let's label:
- AB = height of the tower = 90 m
- BP = distance from the base of the tower (B) to point P = x1 m
- BQ = distance from the base of the tower (B) to point Q = x2 m

### Step 3: Use trigonometric ratios
From point A, we can use the tangent function for both angles of depression:
1. For angle 60 degrees:
\[ \tan(60^\circ) = \frac{AB}{BP} = \frac{90}{x1} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{90}{x1} \implies x1 = \frac{90}{\sqrt{3}} = 30\sqrt{3} \text{ m} \]

2. For angle 30 degrees:
\[ \tan(30^\circ) = \frac{AB}{BQ} = \frac{90}{x2} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{90}{x2} \implies x2 = 90\sqrt{3} \text{ m} \]

### Step 4: Calculate the total distance between P and Q
The total distance between points P and Q is given by:
\[ PQ = BP + BQ = x1 + x2 = 30\sqrt{3} + 90\sqrt{3} = 120\sqrt{3} \text{ m} \]

### Step 5: Calculate the numerical value
Now, we can calculate the numerical value of \(PQ\):
\[ PQ = 120\sqrt{3} \approx 120 \times 1.732 \approx 207.84 \text{ m} \]

### Step 6: Round to the nearest meter
Thus, the distance between points P and Q, rounded to the nearest meter, is approximately:
\[ \text{Distance} \approx 208 \text{ m} \]

### Summary of Steps:
1. Draw a diagram and label the height and distances.
2. Use trigonometric ratios to find x1 and x2.
3. Add x1 and x2 to find the total distance PQ.
4. Calculate the numerical value and round it.