Video: Using the Cosine Rule to Find an Unknown Length in a Triangle in a Real-Life Context

Two boats leave a harbor in different directions. One travels due west at 30 mph, and the other travels on a bearing of 295Β° at 42 mph. How far apart are they after 90 minutes?

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Video Transcript

Two boats leave a harbor in different directions. One travels due west at 30 miles per hour, and the other travels on a bearing of 295 degrees at 42 miles per hour. How far apart are they after 90 minutes?

We will begin this question by drawing a diagram to show the information given. We will consider the direction going vertically upwards to be north. We are told that the first boat travels due west. This is equivalent to a bearing of 270 degrees. The second boat travels on a bearing of 295 degrees. As bearing is a measure clockwise from north, boat two will travel in the direction shown. We know that the first boat travels at a speed of 30 miles per hour. We are interested in where the boat is after 90 minutes. As there are 60 minutes in one hour, 90 minutes is equal to one and a half or 1.5 hours.

Boat two, on the other hand, was traveling at 42 miles per hour. We need to calculate its position also after one and a half hours. We recall that speed is equal to distance divided by time. This means that we can calculate distance by multiplying speed by time. The distance traveled by boat one can be calculated by multiplying 30 by 1.5. This is equal to 45 miles. For the second boat, we can multiply 42 by 1.5. This is equal to 63 miles. We can mark these onto our original diagram as shown. We can then create a triangle with an angle of 25 degrees as 295 minus 270 is 25. The angle between the directions the boats are traveling is 25 degrees.

We now need to calculate how far apart the two boats are after 90 minutes. We can do this by using the cosine rule, which states that π‘Ž squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by cos 𝐴. The length of the triangle we’re trying to calculate is opposite the given angle. Substituting in our values, we get π‘₯ squared is equal to 63 squared plus 45 squared minus two multiplied by 63 multiplied by 45 multiplied by cos 25. Typing in the right-hand side ensuring that our calculator is in degrees gives us 855.2348 and so on.

We can then square root both sides of this equation. This gives us π‘₯ is equal to 29.2443 and so on. We can round this to one decimal place by looking at the number in the hundredths column. As this is less than five, we will round down. We can therefore conclude that the two boats are 29.2 miles apart after 90 minutes.

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