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.
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To solve this problem, we can use trigonometry and the concept of vector addition. Let's start by drawing a diagram to visualize the situation:
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A (8 km/hr)
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B (12 km/hr)
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In this diagram, points A and B represent the starting positions of the two mountain bike riders. The paths they take diverge at a 48-degree angle (as indicated by the lines). Rider A is traveling at 8 km/hr, while rider B is traveling at 12 km/hr.
To find the distance between the two riders after 2 hours, let's calculate the displacements of each rider. Since they have been riding for 2 hours, the displacements will be:
Displacement of rider A = Speed of rider A * Time = 8 km/hr * 2 hr = 16 km
Displacement of rider B = Speed of rider B * Time = 12 km/hr * 2 hr = 24 km
Note: Displacement is a vector quantity that represents the change in position, including both magnitude and direction.
Now we have the displacements of both riders. To find the distance between them, we need to add these displacements as vectors. Since the paths diverge at a 48-degree angle, we'll use vector addition to find the resultant displacement.
Using trigonometry, we find that the horizontal component of the resultant vector is given by:
Horizontal component = (Displacement of A) * cos(48 degrees)
Similarly, the vertical component of the resultant vector is given by:
Vertical component = (Displacement of B) - (Displacement of A) * sin(48 degrees)
Using these formulas, we can calculate the horizontal and vertical components:
Horizontal component = 16 km * cos(48 degrees) ≈ 10.84 km
Vertical component = 24 km - 16 km * sin(48 degrees) ≈ 12.86 km
Finally, we can find the distance between the two riders (which is the magnitude of the resultant displacement) using the Pythagorean theorem:
Distance = √[(Horizontal component)^2 + (Vertical component)^2]
= √[(10.84 km)^2 + (12.86 km)^2]
≈ √[117.6656 km^2 + 165.6196 km^2]
≈ √283.2852 km^2
≈ 16.83 km
Therefore, the two riders are approximately 16.83 km apart after 2 hours.