A truck moved 150 kilometres due east and then 200 kilometres at a direction of 60 degrees. Determine the truck’s displacement, giving its magnitude to the nearest kilometre and its direction to the nearest minute.
If we consider the start point 𝐴, the truck initially moved 150 kilometres due east. It then traveled 200 kilometres at a direction of 60 degrees at which time it arrived at point 𝐵. The magnitude of the truck’s displacement is denoted by 𝑥 and its direction is 𝜃. As angles on a straight line add up to 180 degrees, we can see that the angle inside the triangle is 120 degrees. As 180 minus 60 is 120.
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. 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. Typing this into the calculator gives us a value of 𝑥 squared of 92500. Square rooting both sides of this equation gives us 𝑥 is equal to 304.138. This means that the magnitude of the truck’s displacement is 304 kilometres to the nearest kilometre.
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. Substituting our values into this equation gives us 200 divided by sin 𝜃 is equal to 304 divided by sin 120. We can rearrange this equation so that sin 𝜃 is equal to sin of 120 divided by 304 multiplied by 200.
To calculate the angle 𝜃, we can do sin to the minus one or inverse sin of 0.569. This is equal to 34.715. We were asked to give our answer to the nearest minute. Therefore, we need to convert or change 0.715 into minutes. 0.715 multiplied by 60 is 42.9. As this is 43 minutes to the nearest minute, our angle 𝜃 is 34 degrees and 43 minutes.
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.