Question
What is the height of point X above the level of point Y given the angle of elevation of 30 degrees and a distance of 40 meters between the two points?
Answers
Answered by
GPT-4o mini
To determine the height of point X above point Y given an angle of elevation of 30 degrees and a horizontal distance of 40 meters, we can use some basic trigonometry.
In this case, we can use the tangent function, which relates the angle of elevation to the opposite side (the height difference) and the adjacent side (the horizontal distance):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \(\theta = 30\) degrees
- \(\text{opposite} = \text{height of point X above point Y}\)
- \(\text{adjacent} = 40\) meters
Rearranging the formula to find the height:
\[
\text{height} = \tan(\theta) \times \text{adjacent}
\]
Substituting in the values:
\[
\text{height} = \tan(30^\circ) \times 40 \text{ meters}
\]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577\):
\[
\text{height} = \frac{1}{\sqrt{3}} \times 40 \approx 0.577 \times 40 \approx 23.09 \text{ meters}
\]
Thus, the height of point X above point Y is approximately **23.09 meters**.
In this case, we can use the tangent function, which relates the angle of elevation to the opposite side (the height difference) and the adjacent side (the horizontal distance):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Where:
- \(\theta = 30\) degrees
- \(\text{opposite} = \text{height of point X above point Y}\)
- \(\text{adjacent} = 40\) meters
Rearranging the formula to find the height:
\[
\text{height} = \tan(\theta) \times \text{adjacent}
\]
Substituting in the values:
\[
\text{height} = \tan(30^\circ) \times 40 \text{ meters}
\]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577\):
\[
\text{height} = \frac{1}{\sqrt{3}} \times 40 \approx 0.577 \times 40 \approx 23.09 \text{ meters}
\]
Thus, the height of point X above point Y is approximately **23.09 meters**.