Houses Are Arranged In A School Compound .Thus Blue House Is 4Km Due East Of Red ,Yellow Is 3Km Due South Of Blue And Green Is 4Km On A Bearing Of 230Degrees From The Yellow . Calculate The Distance And Bearing Of The Red House From Green House
To calculate the distance and bearing of the Red house from the Green house in the given scenario, we can use the concept of vector addition and trigonometry.
First, let's draw a diagram to visualize the arrangement of the houses:
```
Red
|
______|______ Blue
|
Yellow
|
Green
```
Given information:
1. Blue House is 4 km due east of Red House.
2. Yellow House is 3 km due south of Blue House.
3. Green House is 4 km on a bearing of 230 degrees from Yellow House.
Now, let's calculate the position of each house relative to a common reference point (we'll take Red House as the reference).
Let's consider the following distances and directions:
- Red to Blue (RB): 4 km (due east)
- Blue to Yellow (BY): 3 km (due south)
- Yellow to Green (YG): 4 km (230 degrees)
Step 1: Determine the position of each house relative to the reference point (Red House).
- Blue House: RB (due east)
- Yellow House: RB + BY (due east + due south)
- Green House: RB + BY + YG (due east + due south + 230 degrees)
Step 2: Calculate the distance between Green and Red Houses using the Pythagorean theorem.
- Distance (GR) = sqrt((RB + BY + YG)^2 + BY^2)
Substituting the given values:
GR = sqrt((4 + 3 + 4)^2 + 3^2)
= sqrt(11^2 + 9)
= sqrt(121 + 9)
= sqrt(130)
≈ 11.40 km (rounded to two decimal places)
Step 3: Calculate the bearing of the Green House from the Red House using trigonometry.
- Bearing = arctan(BY / (RB + BY + YG))
Substituting the given values:
Bearing = arctan(3 / (4 + 3 + 4))
= arctan(3 / 11)
≈ 15.87 degrees (rounded to two decimal places)
Therefore, the distance from the Red House to the Green House is approximately 11.40 km, and the bearing is approximately 15.87 degrees.