Two paths diverge at a 48 degree angle. Two mountain bike riders take separate routes at 8km/hr and 12km/hr. How far apart are they after 2 hours? Include a diagram.
Is elevation constant?
distance=velocty*time
you do not have time.
Figure the distance for each. You have on the triangle SAS
you can use the law of consines to figure the last side.
distance^2=d2^2+d1^2 -2*d2*d1*cos48
To find out how far apart the two mountain bike riders are after 2 hours, we can use the concept of vectors and trigonometry. Let's break down the problem step by step and include a diagram to make it easier to understand.
Step 1: Draw a diagram
To visualize the situation, draw a diagram with two paths diverging at a 48-degree angle. Label one path as A and the other path as B. Place two points on each path to represent the starting positions of the two mountain bike riders. Also, mark the angle between the two paths as 48 degrees.
```
A
|\
| \
| \
| \
| \
___|_____\
B
```
Step 2: Calculate the distance each rider has traveled
We are given that the first rider travels at a speed of 8 km/hr and the second rider at 12 km/hr. After 2 hours, each rider will have traveled a certain distance.
Rider A: Distance = Speed × Time = 8 km/hr × 2 hr = 16 km
Rider B: Distance = Speed × Time = 12 km/hr × 2 hr = 24 km
Step 3: Use trigonometry to find the distance between the two riders
To find the distance between the two riders after 2 hours, we need to calculate the horizontal and vertical components of their positions.
In the diagram, mark the horizontal line connecting the starting positions of the riders as "dH" and the vertical line as "dV".
```
A
|\
| \
dV | \
| \
___|____\dH
B
```
Since we have a right triangle formed by dH, dV, and the angle of divergence (48 degrees), we can use trigonometry.
Step 4: Calculate dH and dV using trigonometric functions
We can use the tangent function to relate the angle and the sides of the triangle.
tan(48 degrees) = dV / dH
Rearranging the formula, we get:
dV = dH * tan(48 degrees)
Step 5: Substitute the values to calculate the distances
Substituting the calculated distances for dH, we can determine the vertical distance between the two riders.
dV = 16 km * tan(48 degrees) ≈ 16 km * 1.108 ≈ 17.73 km
Therefore, after 2 hours, the two mountain bike riders are approximately 17.73 kilometers apart.
Please note that this calculation assumes the riders maintain a constant speed and do not deviate from their paths while riding.