To find the height of the hill given the angle of depression and the horizontal distance from the point to the foot of the hill, we can use trigonometric principles. Here is a step-by-step solution:
1. Convert the angle from degrees and minutes to decimal degrees:
The angle of depression provided is [tex]\(61^\circ 48'\)[/tex].
To convert this angle to decimal degrees:
- The degrees part remains the same: 61 degrees.
- There are 60 minutes in a degree, so we convert 48 minutes to degrees by dividing 48 by 60.
[tex]\[
\text{Decimal degrees} = 61 + \frac{48}{60}
\][/tex]
Simplifying, we find:
[tex]\[
\text{Decimal degrees} = 61 + 0.8 = 61.8^\circ
\][/tex]
2. Determine the horizontal distance:
The distance from the point X to the foot of the hill is given as 35 meters.
3. Use trigonometric relationships:
Since the angle given is the angle of depression, it is equivalent to the angle of elevation from point X to the top of the hill (due to the properties of alternate interior angles created by a transversal cutting parallel lines).
We will use the tangent function, which relates the angle of elevation to the height of the hill and the horizontal distance:
[tex]\[
\tan(\theta) = \frac{\text{height}}{\text{distance from foot}}
\][/tex]
Substituting the given values:
[tex]\[
\tan(61.8^\circ) = \frac{\text{height}}{35}
\][/tex]
4. Solve for the height:
To find the height, we rearrange the equation:
[tex]\[
\text{height} = 35 \times \tan(61.8^\circ)
\][/tex]
5. Calculate the height using the tangent of the angle:
Using the trigonometric value of [tex]\(\tan(61.8^\circ)\)[/tex], we find the height:
[tex]\[
\text{height} \approx 35 \times 1.865 = 65.27 \text{ meters}
\][/tex]
Therefore, the height of the hill is approximately 65.27 meters.
