WEBVTT
00:00:01.630 --> 00:00:09.750
A truck moved 150 kilometres due east and then 200 kilometres at a direction of 60 degrees.
00:00:10.460 --> 00:00:17.960
Determine the truck’s displacement, giving its magnitude to the nearest kilometre and its direction to the nearest minute.
00:00:19.950 --> 00:00:26.350
If we consider the start point 𝐴, the truck initially moved 150 kilometres due east.
00:00:27.650 --> 00:00:36.860
It then traveled 200 kilometres at a direction of 60 degrees at which time it arrived at point 𝐵.
00:00:38.670 --> 00:00:45.010
The magnitude of the truck’s displacement is denoted by 𝑥 and its direction is 𝜃.
00:00:47.030 --> 00:00:55.210
As angles on a straight line add up to 180 degrees, we can see that the angle inside the triangle is 120 degrees.
00:00:55.550 --> 00:00:59.610
As 180 minus 60 is 120.
00:01:01.530 --> 00:01:10.400
In order to calculate 𝑥, we can use the cosine rule: 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏 𝑐 cos 𝐴.
00:01:12.630 --> 00:01:30.200
Substituting in the values from the diagram gives us 𝑥 squared is equal to 150 squared plus 200 squared minus two multiplied by 150 multiplied by 200 multiplied by cos of 120 degrees.
00:01:31.750 --> 00:01:37.710
Typing this into the calculator gives us a value of 𝑥 squared of 92500.
00:01:38.970 --> 00:01:45.710
Square rooting both sides of this equation gives us 𝑥 is equal to 304.138.
00:01:47.120 --> 00:01:53.890
This means that the magnitude of the truck’s displacement is 304 kilometres to the nearest kilometre.
00:01:55.300 --> 00:02:05.660
In order to calculate the angle 𝜃, the direction of the truck, we’ll use the sine rule: 𝑎 divided by sin 𝐴 is equal to 𝑏 divided by sin 𝐵.
00:02:07.170 --> 00:02:16.980
Substituting our values into this equation gives us 200 divided by sin 𝜃 is equal to 304 divided by sin 120.
00:02:18.380 --> 00:02:28.410
We can rearrange this equation so that sin 𝜃 is equal to sin of 120 divided by 304 multiplied by 200.
00:02:29.780 --> 00:02:37.080
To calculate the angle 𝜃, we can do sin to the minus one or inverse sin of 0.569.
00:02:38.300 --> 00:02:42.030
This is equal to 34.715.
00:02:43.310 --> 00:02:46.110
We were asked to give our answer to the nearest minute.
00:02:46.560 --> 00:02:51.890
Therefore, we need to convert or change 0.715 into minutes.
00:02:53.120 --> 00:02:57.960
0.715 multiplied by 60 is 42.9.
00:02:59.080 --> 00:03:06.970
As this is 43 minutes to the nearest minute, our angle 𝜃 is 34 degrees and 43 minutes.
00:03:08.330 --> 00:03:34.210
We can therefore say that a truck that is moved 150 kilometres due east and then 200 kilometres at a direction of 60 degrees has a displacement with magnitude 304 kilometres and a direction 34 degrees and 43 minutes north of east or 34 degrees and 43 minutes from the horizontal.