PQ is a tower standing on horizontal plane, Q being its foot. Two points A and B are taken on the plane such that AB=21 and (c) QAB is a right angle. It is found that cot PAQ=2/5 and cot PBQ=3/5, then the height of tower is ____.

Answer

To find the height of the tower PQ, we will follow these steps:

### Step 1: Understand the given information
We have a tower PQ with Q as its foot on the horizontal plane. Points A and B are on the plane such that AB = 21 and ∠QAB = 90°. We know that cot PAQ = 2/5 and cot PBQ = 3/5.

### Step 2: Set up the triangles
Let:
- H = height of the tower PQ
- AQ = distance from point A to point Q
- BQ = distance from point B to point Q

From the definitions of cotangent:
- cot PAQ = AQ / H = 2/5 → (1)
- cot PBQ = BQ / H = 3/5 → (2)

### Step 3: Express AQ and BQ in terms of H
From equation (1):
\[ AQ = H \cdot \frac{2}{5} \]
From equation (2):
\[ BQ = H \cdot \frac{3}{5} \]

### Step 4: Use the Pythagorean theorem in triangle ABQ
Since ∠QAB is a right angle, we can apply the Pythagorean theorem:
\[ AB^2 = AQ^2 + BQ^2 \]

Substituting the values we found:
\[ 21^2 = \left(H \cdot \frac{2}{5}\right)^2 + \left(H \cdot \frac{3}{5}\right)^2 \]

### Step 5: Simplify the equation
Calculating the squares:
\[ 441 = \left(\frac{4H^2}{25}\right) + \left(\frac{9H^2}{25}\right) \]
\[ 441 = \frac{4H^2 + 9H^2}{25} \]
\[ 441 = \frac{13H^2}{25} \]

### Step 6: Solve for H^2
Multiply both sides by 25:
\[ 441 \cdot 25 = 13H^2 \]
\[ 11025 = 13H^2 \]

Now, divide by 13:
\[ H^2 = \frac{11025}{13} \]
\[ H^2 = 850.3846 \]

### Step 7: Calculate H
Taking the square root:
\[ H = \sqrt{850.3846} \]
\[ H \approx 29.2 \]

### Step 8: Final answer
The height of the tower PQ is approximately 29.2 meters.

---