Two bike riders are travelling down a road when they reach a split in the road. The two bikers take separate routes at 17 km/h and 24 km/h. About 2 hours later they are 38 km apart. Draw a labeled sketch of the situation and then determine the angle between the path of the two bikers to the nearest degree.
To solve this problem, we first need to understand the situation and gather the necessary information. Let's start by drawing a labeled sketch of the situation.
A
------>
17 km/h
B ------------ C
<-------
24 km/h
In the sketch above, we have two bikers, A and B, traveling down the road. They reach a split in the road and take separate routes. Biker A travels at a speed of 17 km/h, while Biker B travels at a speed of 24 km/h. We need to find the angle between the paths of the two bikers when they are 38 km apart, approximately 2 hours later.
To find the angle between the paths of the two bikers, we need to consider the relative distances they cover in those 2 hours. Biker A covers a distance of 17 km/h * 2 h = 34 km, and Biker B covers a distance of 24 km/h * 2 h = 48 km.
Now that we know the distances each biker covers, we can label the points on the sketch accordingly.
A
------>
34 km
B ------------ C
<-------
48 km
The angle we need to find is the angle BAC. To find this angle, we can use the Law of Cosines. The Law of Cosines states that in any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our scenario, biker A has covered a distance of 34 km, and biker B has covered a distance of 48 km. The total distance between them is 38 km. Therefore, we have:
c = 38 km
a = 34 km
b = 48 km
Substituting these values into the Law of Cosines equation, we can solve for cos(C):
38^2 = 34^2 + 48^2 - 2 * 34 * 48 * cos(C)
Simplifying the equation, we have:
1444 = 1156 + 2304 - 3264 * cos(C)
Now we solve for cos(C):
3264 * cos(C) = 2304 + 1156 - 1444
cos(C) = (2304 + 1156 - 1444) / 3264
cos(C) = 0.310
To find the angle C, we can take the inverse cosine (arccos) of cos(C) using a calculator:
C ≈ arccos(0.310)
C ≈ 71.97 degrees
Therefore, the angle between the paths of the two bikers is approximately 72 degrees to the nearest degree.