The angle of elevation of the top of a tower at a point A due south of the tower is `alpha` and at `beta` due east of the tower is `beta`. If AB=d, then height of the tower is

Options

A. `dsqrt(cot^(2)alpha + cot^(2)beta)`
B. `d/sqrt(cot^(2)alpha + cot^(2)beta)`
C. `d cot alpha`
D. None of these

Answer

To find the height of the tower given the angles of elevation and the distance between the points, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Let the height of the tower be \( h \).
- The angle of elevation from point A (due south) is \( \alpha \).
- The angle of elevation from point B (due east) is \( \beta \).
- The distance \( AB = d \).

2. **Set Up the Right Triangles**:
- From point A, we can form a right triangle where:
- The opposite side (height of the tower) is \( h \).
- The adjacent side (distance from the base of the tower to point A) is \( OA \).
- From point B, we can form another right triangle where:
- The opposite side (height of the tower) is \( h \).
- The adjacent side (distance from the base of the tower to point B) is \( OB \).

3. **Apply the Tangent Function**:
- For triangle AOP:
\[ \tan(\alpha) = \frac{h}{OA} \implies OA = \frac{h}{\tan(\alpha)} = h \cdot \cot(\alpha) \]
- For triangle BOP:
\[ \tan(\beta) = \frac{h}{OB} \implies OB = \frac{h}{\tan(\beta)} = h \cdot \cot(\beta) \]

4. **Use the Pythagorean Theorem**:
- In triangle AOB, we can apply the Pythagorean theorem:
\[ AB^2 = OA^2 + OB^2 \]
- Substituting the values of \( OA \) and \( OB \):
\[ d^2 = (h \cdot \cot(\alpha))^2 + (h \cdot \cot(\beta))^2 \]
- This simplifies to:
\[ d^2 = h^2 \cdot \cot^2(\alpha) + h^2 \cdot \cot^2(\beta) \]
- Factoring out \( h^2 \):
\[ d^2 = h^2 (\cot^2(\alpha) + \cot^2(\beta)) \]

5. **Solve for the Height \( h \)**:
- Rearranging the equation gives:
\[ h^2 = \frac{d^2}{\cot^2(\alpha) + \cot^2(\beta)} \]
- Taking the square root:
\[ h = \frac{d}{\sqrt{\cot^2(\alpha) + \cot^2(\beta)}} \]

### Final Answer:
Thus, the height of the tower is:
\[ h = \frac{d}{\sqrt{\cot^2(\alpha) + \cot^2(\beta)}} \]