To solve for the distance between the tree the observer is on and the house, we can use trigonometry. Since the observer is looking at the top of the house from the top of the tree, we can form a right-angled triangle with the height difference between the tree and the house as one leg and the distance between the two as the other leg.
Here is a step-by-step solution:
1. Find the vertical height difference between the observer and the top of the house.
The height of the tree is 100 meters, and the height of the house is 20 meters.
The difference in height is 100 meters - 20 meters = 80 meters.
2. Since the observer can see the top of the house, the angle of depression corresponds to the angle of elevation from the base of the tree to the top of the house. We use the angle of elevation because it is inside the right triangle we are considering.
3. However, suppose we do not have the value of the angle of depression/elevation. We cannot directly calculate the distance without it. In trigonometry, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
We would use the formula:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
where \(\theta\) is the angle of elevation and the opposite side is the height difference between the tree top and the house top, and the adjacent side is what we're looking for – the distance between the house and the tree.
4. We cannot solve this equation without knowing the angle of elevation (or angle of depression, which would be the same angle). If this angle were provided, let's call it \( \alpha \), then we could rearrange the formula to:
\[ \text{distance} = \frac{\text{opposite}}{\tan(\alpha)} \]
5. Substituting the known values:
\[ \text{distance} = \frac{80 \text{ meters}}{\tan(\alpha)} \]
6. Without the angle \(\alpha\), we cannot proceed with the calculation.
In conclusion, we need the angle of elevation (or depression) to calculate the distance between the tree and the house. If you can provide this angle, we can then find the distance using the tangent function as shown above.