Two paths diverge at a 44 degree angle. Two mountain bike riders take separate routes at 6.5 km/ hr and 10 km/ hr. How far apart are they after 2 hours? Include a diagram.
They are each 13 and 20 km from the starting point, at an angle of 44 degrees.
Solve the triangle by the cosine rule which gives the distance between the two:
c²=a²+b²-2ab cos(C)
where
c=distance between bikers
a=13km
b=20km
C=44°
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13.962km apart
To determine the distance between the two mountain bike riders after 2 hours, we need to use trigonometry.
First, let's draw a diagram to visualize the situation. We have two paths that diverge at a 44-degree angle. Let's label this angle as θ.
/\
/ \
D / \ E
/ \
/________\
P Q
In this diagram, P and Q represent the starting points of the two mountain bike riders, and D and E represent their positions after 2 hours. We want to find the distance between D and E, which is represented by the segment DE.
To calculate the distance between D and E, we can break it down into two components: the horizontal distance, called the x-component, and the vertical distance, called the y-component.
The x-component is the distance covered by the rider traveling at 10 km/hr, and the y-component is the distance covered by the rider traveling at 6.5 km/hr.
The x-component can be calculated as follows:
x-component = (speed of the rider at 10 km/hr) * (time of 2 hours) * (cosine of the angle θ)
The y-component can be calculated as follows:
y-component = (speed of the rider at 6.5 km/hr) * (time of 2 hours) * (sine of the angle θ)
Now, let's plug in the values and calculate the x-component and y-component separately.
For the x-component:
speed of the rider at 10 km/hr = 10 km/hr
time = 2 hours
angle θ = 44 degrees
x-component = 10 km/hr * 2 hours * cosine(44 degrees)
x-component ≈ 10 * 2 * 0.7193 ≈ 14.386 km
For the y-component:
speed of the rider at 6.5 km/hr = 6.5 km/hr
time = 2 hours
angle θ = 44 degrees
y-component = 6.5 km/hr * 2 hours * sine(44 degrees)
y-component ≈ 6.5 * 2 * 0.6947 ≈ 9.0235 km
Using the Pythagorean theorem, we can find the distance between D and E:
distance = square root(x-component^2 + y-component^2)
distance ≈ square root(14.386^2 + 9.0235^2)
distance ≈ square root(206.559 + 81.421)
distance ≈ square root(288.98)
distance ≈ 17.01 km
Therefore, the two mountain bike riders are approximately 17.01 kilometers apart after 2 hours.