From the top of a hill, the angles of depression of two consecutive kilometre stones, due east, are found to be `30^(@) and 45^(@)` respectively. Find the distance of the two from the foot of the hill

Answer

To solve the problem step by step, we will use trigonometric principles and the properties of right triangles.

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Let the height of the hill be \( AB = h \).
- Let \( C \) be the position of the first kilometer stone (at an angle of depression of \( 30^\circ \)).
- Let \( D \) be the position of the second kilometer stone (at an angle of depression of \( 45^\circ \)).
- The distance from the foot of the hill to stone \( C \) is \( x \) kilometers, and the distance to stone \( D \) is \( x + 1 \) kilometers (since the stones are consecutive).

2. **Use the Angle of Depression**:
- For angle \( 45^\circ \) (stone \( D \)):
\[ \tan(45^\circ) = \frac{h}{x} \]
Since \( \tan(45^\circ) = 1 \):
\[ h = x \quad \text{(Equation 1)} \]

- For angle \( 30^\circ \) (stone \( C \)):
\[ \tan(30^\circ) = \frac{h}{x + 1} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{x + 1} \]
Cross-multiplying gives:
\[ h \sqrt{3} = x + 1 \quad \text{(Equation 2)} \]

3. **Substitute Equation 1 into Equation 2**:
- Substitute \( h = x \) into Equation 2:
\[ x \sqrt{3} = x + 1 \]
- Rearranging gives:
\[ x \sqrt{3} - x = 1 \]
\[ x(\sqrt{3} - 1) = 1 \]
\[ x = \frac{1}{\sqrt{3} - 1} \]

4. **Rationalize the Denominator**:
- Multiply the numerator and denominator by the conjugate \( \sqrt{3} + 1 \):
\[ x = \frac{1(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{\sqrt{3} + 1}{3 - 1} = \frac{\sqrt{3} + 1}{2} \]

5. **Calculate the Value of \( x \)**:
- Now, substituting the value of \( \sqrt{3} \approx 1.732 \):
\[ x = \frac{1.732 + 1}{2} = \frac{2.732}{2} \approx 1.366 \text{ kilometers} \]

6. **Find the Distance to the Second Stone**:
- The distance to stone \( D \) is:
\[ x + 1 = 1.366 + 1 = 2.366 \text{ kilometers} \]

### Final Answers:
- Distance to the first stone \( C \): \( 1.366 \) kilometers
- Distance to the second stone \( D \): \( 2.366 \) kilometers