Question Video: Using the Cosine Rule to Solve Word Problems Mathematics

A biker traveled from city 𝐴 to city 𝐡 via city 𝐢 with a uniform speed of 52 km/h. He then returned to city 𝐴 with a uniform speed of 89 km/h. Find, in minutes, the total time of his whole journey to two decimal places.

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Video Transcript

A biker traveled from city 𝐴 to city 𝐡 via city 𝐢 with a uniform speed of 52 kilometers per hour. He then returned to city 𝐴 with a uniform speed of 89 kilometers per hour. Find, in minutes, the total time of his whole journey to two decimal places.

We are told that the biker initially travels from city 𝐴 to city 𝐡 via city 𝐢 as shown. He then returns directly to city 𝐴. We are asked to calculate the total time for the entire journey. We can do this using our speed–distance–time triangle, where the time can be calculated by dividing the distance by the speed.

We are given the speeds in kilometers per hour. The distance is measured in kilometers. Therefore, the time will be in hours. We will begin by calculating the direct distance from city 𝐡 to city 𝐴. We will call this π‘₯ kilometers.

Given the lengths of two sides as well as the included angle of any triangle, we can calculate the length of the third side. We do this using the law of cosines, otherwise known as the cosine rule. This states that the length 𝑐 squared is equal to π‘Ž squared plus 𝑏 squared minus two π‘Žπ‘ multiplied by the cos of angle 𝐢. Substituting in the values in this question, we get π‘₯ squared is equal to 19 squared plus 15 squared minus two multiplied by 19 multiplied by 15 multiplied by the cos of 88 degrees.

Typing the right-hand side of the equation into our calculator gives us 566.107 and so on. We can then square root both sides of this equation, giving us π‘₯ is equal to 23.793 and so on. The direct distance between city 𝐡 and city 𝐴 is 23.793 kilometers. We will now calculate the time taken for each part of the journey.

In the first part of the journey, the biker traveled a total of 34 kilometers, as 15 plus 19 is equal to 34. We know that his speed doing this part of the journey was 52 kilometers per hour. This means we can calculate the time by dividing 34 by 52. This gives us 0.6538 and so on hours. In the second part of the journey, the biker traveled a distance of 23.793 kilometers. He covered this distance at a uniform speed of 89 kilometers per hour. We can divide these values to calculate the time taken.

This is equal to 0.2673 and so on hours. We can calculate the total time by adding 0.6538 and 0.2673. This is equal to 0.9211. This time is given in hours, and we want to give our answer in minutes. As there are 60 minutes in one hour, we need to multiply this answer by 60. This is equal to 55.266. And rounding to two decimal places, we get an answer of 55.27. The total time for the biker’s whole journey is 55.27 minutes.

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