A motor cruising south along the highway is 16 km from the nearest town. After travelling a distance of 32 km, he finds that he is 19 km from the same town. Find the bearing of the town from the second position of the motorist.
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To find the bearing of the town from the second position of the motorist, we need to first understand the concept of bearing.
Bearing is the direction from a particular point (usually the starting point) to another point, measured in degrees clockwise from north. In this case, the starting point is the second position of the motorist, and we want to determine the bearing of the town relative to that position.
To visualize this situation, let's create a diagram:
T (Town)
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| x (Second Position of Motorist)
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M (Motorist's First Position)
In this diagram, the letters T and M represent the town and the first position of the motorist, respectively. The symbol x represents the second position of the motorist.
Now, let's analyze the information given.
The motorist is 16 km from the nearest town (M -> T), and after traveling a distance of 32 km, he is 19 km from the same town (M -> x -> T).
We can use trigonometry to determine the bearing of the town from the second position of the motorist.
Let's define:
a = Distance from the second position of the motorist to the town (x -> T)
b = Distance from the first position of the motorist to the town (M -> T)
c = Distance between the first and second positions of the motorist (M -> x)
Now, let's apply the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, angle C is the angle between the line connecting the first and second positions of the motorist (M -> x) and the line connecting the first position of the motorist to the town (M -> T).
By substituting the values into the equation, we have:
32^2 = 19^2 + 16^2 - 2 * 19 * 16 * cos(C)
Simplifying the equation will lead us to the value of cos(C).