Alessandra and Brigitte both start driving away from a train station at the same time. They both drive at 30 miles per hour. There is also a train track from the station to a distant town, 𝑇. The train track is on a bearing of 054 degrees from 𝑆. Alessandra drives on a bearing of 118 degrees from 𝑆. Brigitte drives on a bearing of 347 degrees from 𝑆. Part a), how long does it take Brigitte to drive 10 miles? There is also a part b) and c) that we will look at later.
The information that we’re given about the bearings is not relevant to part a). However, it is worth putting this on the diagram at the start. Bearings are given with three figures, which is why we have 054 and not just 54 degrees. They are also measured clockwise from north. As the train track is on a bearing of 054 degrees from 𝑆, we know that the angle between our north line on the line 𝑇 is 54 degrees. We’re also told that Alessandra drives on a bearing of 118 degrees. This means that the angle between the north line and line 𝐴 is 118 degrees. We can, therefore, calculate the angle between line 𝑇 and line 𝐴 by subtracting 54 from 118. This is equal to 64 degrees.
Finally, we’re told that Brigitte drives on a bearing of 347 degrees from 𝑆. Angles in a circle, or angles at a point, add up to 360 degrees. This means that we can calculate the anticlockwise angle between the north line and line 𝐵 by subtracting 347 from 360. This is equal to 13 degrees. The angle between line 𝐵 and the north line is 13 degrees. We will use these angles that we have worked out in part b) and c). Part a) asks us to calculate how long it takes Brigitte to drive 10 miles. We know that she’s driving at a speed of 30 miles per hour. The distance she has traveled is 10 miles. We want to calculate the time that this will take. The easiest way to do this is to use our speed-distance-time triangle.
We cover up the letter we’re trying to work out, in this case time. We’re left with distance over speed. Therefore, the time can be calculated by dividing the distance by the speed. The distance was 10 and the speed was 30. Therefore, the time is 10 divided by 30. Both of these numbers can be divided by 10. 10 divided by 10 is equal to one, and 30 divided by 10 is equal to three. Remember, with fractions whatever you do to the numerator you must do to the denominator. We can, therefore, say that the time it takes for Brigitte to drive 10 miles is one-third of an hour.
We were asked to give our answer in minutes. And we know that 60 minutes is equal to one hour. Multiplying one-third by 60 or finding a third of 60 will work out the number of minutes it takes Brigitte to drive 10 miles. One-third multiplied by 60 is equal to 20. So it takes Brigitte 20 minutes to drive 10 miles. An alternative method here would be to scale up and down. We’re told that Brigitte drives at 30 miles per hour. This means that she travels 30 miles in 60 minutes. We wanted to work out how long it took her to drive 10 miles. 30 divided by three is equal to 10. Therefore, we need to divide the time by three. 60 divided by three is equal to 20. So once again, we have proved that she drives 10 miles in 20 minutes.
The second part of the question says the following. b) Brigitte says, after one hour, Alessandra has driven 30 miles. The shortest distance from Alessandra to the train track is, therefore, 30 miles. Is Brigitte correct? Tick one box and give reasons for your answer.
As we have already put the information about the bearings on the diagram, we will clear this to make some space. Let’s firstly consider the angle between the direction of the train track, 𝑇, and the direction of Alessandra, 𝐴. The angle between these two lines is 64 degrees. The shortest distance between any two straight lines is always a right angle. This can be shown in pink. If we create a right angle triangle, we can see that the distance that Alessandra has travelled was 30 miles. This is the hypotenuse or longest side of the triangle as it is opposite the right angle. The distance 𝑥 between Alessandra and the train track will actually be less than 30 miles, as the other two sides of any right angled triangle are less than the hypotenuse.
We can work this length or distance out using trigonometry. As the triangle is right angled, we can use SOHCAHTOA. We know that the hypotenuse is 30. And we’re trying to work out the opposite. Substituting in the values from our triangle gives us sin of 64 is equal to 𝑥 divided by 30, the opposite divided by the hypotenuse. Multiplying both sides of this equation by 30 gives us 30 multiplied by sin 64 is equal to 𝑥. Typing this into our calculator gives us a value for 𝑥 of 26.96 to two decimal places. We can, therefore, say that the shortest distance between Alessandra and the train track is just under 27 miles. We can therefore conclude that Brigitte is not correct. This would only be true if Alessandra was traveling at an angle of 90 degrees from the train track. As the angle between the train track and Alessandra is 64 degrees, the distance will be less than 30 miles.
The final part of the question says the following. c) After one hour, who is farthest from the nearest point on the train track? Tick one box and show your working. Is it Alessandra, Brigitte, or are they both the same?
We have already worked out that Alessandra was 26.96 miles away from the train track. This is because the angle between her direction of travel and the train track was 64 degrees. The angle between Brigitte’s direction of travel and the train track is equal to 13 degrees plus 54 degrees. 13 plus 54 is equal to 67. Therefore, the angle between Brigitte and the train track was 67 degrees. We can use right angle trigonometry or SOHCAHTOA once again to calculate the distance from Brigitte to the train track. As Brigitte is traveling the same speed as Alessandra, she will also have traveled 30 miles. The angle between her and the train track is 67 degrees. We need to calculate the distance, 𝑦, the shortest distance between Brigitte and the train track.
Once again, we’re dealing with the opposite and the hypotenuse. This time, we have sin of 67 degrees is equal to 𝑦 divided by 30. Multiplying both sides of the equation by 30 gives us 30 multiplied by sin 67 is equal to 𝑦. Typing this into our calculator gives us an answer to two decimal places of 27.62. The shortest distance from Brigitte to the train track after one hour is 27.62 miles. We were asked to work out who is farther from the train track. So the correct answer is Brigitte, as 27.62 is greater than 26.96.
Whilst we can use trigonometry to work out which driver was farthest from the track, we could do so just by looking at the angles. An angle of zero degrees corresponds to traveling along the track, whereas an angle of 90 degrees corresponds to traveling away from the track. The larger the angle up to 90 degrees, the farther the driver is from the closest point on the train track. As 67 degrees is greater than 64 degrees, Brigitte is farther away.