To find the distance between the house and the tree given the angle of elevation, we will use trigonometric principles. Here's the detailed, step-by-step solution:
### Step 1: Understand the Problem
- Height of the house (h_house) = 8 meters
- Height of the tree (h_tree) = 20 meters
- Angle of elevation (θ) from the roof of the house to the top of the tree = 30°
### Step 2: Calculate the Height Difference
First, we need to find the difference in height between the tree and the house.
[tex]\[
\text{Height difference} = h_{\text{tree}} - h_{\text{house}} = 20 \, \text{meters} - 8 \, \text{meters} = 12 \, \text{meters}
\][/tex]
### Step 3: Use Trigonometry to Find the Distance
We will use the tangent function to find the horizontal distance ([tex]\( d \)[/tex]) between the house and the tree. The tangent function relates the angle of elevation to the opposite side (height difference) and the adjacent side (distance between the house and the tree):
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the opposite side is the height difference, and the adjacent side is the distance we need to find.
[tex]\[
\tan(30^\circ) = \frac{12 \, \text{meters}}{d}
\][/tex]
### Step 4: Solve for the Distance
Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{12 \, \text{meters}}{\tan(30^\circ)}
\][/tex]
We know from trigonometric tables or a calculator that:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577
\][/tex]
So,
[tex]\[
d = \frac{12 \, \text{meters}}{0.577} \approx 20.8 \, \text{meters}
\][/tex]
### Conclusion
The distance between the house and the tree is approximately 20.8 meters.
