Final answer:
The height of the cloud above the lake is approximately 86.19 meters, calculated using trigonometry relations of angles of elevation and depression from Dustin's point of view standing on a 40-meter high cliff.
Step-by-step explanation:
To find the height of the cloud above the lake, we use trigonometry. Dustin observes the cloud at an angle of elevation of 30° and the reflection at an angle of depression of 60°. The height of the cliff is 40 meters. Since angles of elevation and depression are measured from the horizontal, the angle between Dustin's line of sight to the cloud and to its reflection is 90° (30° + 60°). The height of the cloud can be found by considering two right-angled triangles that Dustin forms with his line of sight to the cloud and to the reflection.
Step 1:
Triangle ADC represents the direct line of sight to the cloud, with angle CAD being 30°, and side AC the height of the cliff (40 meters).
Step 2:
Triangle BEC represents the line of sight to the reflection of the cloud, with angle CBE being 60°, and side BC the height of the cliff (40 meters).
Step 3:
Using the trigonometric relations:
- Tan 30° = opposite/adjacent → CD/AC → CD = AC * Tan 30°
- Tan 60° = opposite/adjacent → CE/BC → CE = BC * Tan 60°
Step 4:
Since CE and CD form a straight line, the total height of the cloud above the lake is the sum of CD and CE. Using the trigonometry values (Tan 30° = 1/√3, and Tan 60° = √3):
- CD = 40 * 1/√3
- CE = 40 * √3
Step 5:
Therefore, the height of the cloud above the lake, DE = CD + CE = 40 * (1/√3 + √3).
After calculation, DE = 40 * (1/√3 + √3) ≈ 40 * 2.1547 ≈ 86.19 meters.
So, the height of the cloud above the lake is approximately 86.19 meters.