Question Video: Using the Law of Sines to Calculate an Unknown Length in a Real-World Context Involving Angles of Elevation Mathematics

A boat was sailing in a straight line toward a rock with a uniform velocity of 96 m/min. At one point, the angle of elevation to the top of the rock was 39°, and 3 minutes later it became 44°. Find the height of the rock giving the answer to the nearest meter.


Video Transcript

A boat was sailing in a straight line toward a rock with a uniform velocity of 96 meters per minute. At one point, the angle of elevation to the top of the rock was 39 degrees. And three minutes later, it became 44 degrees. Find the height of the rock giving the answer to the nearest meter.

So, firstly, we need to think carefully about how we can draw a diagram to represent this problem. We have a rock and a boat, which we’ll simplify as a pink dot. Initially, we’re told that the angle of elevation to the top of the rock is 39 degrees. Remember, an angle of elevation is measured from the horizontal to the line of sight when we look up towards something. So that’s this angle here. The boat then continues on its journey in a straight line towards the rock. And three minutes later, the new angle of elevation is 44 degrees. So from the new position of the boat, that’s this angle here.

We can see that we have a triangle formed by the two lines of sight from the two positions to the top of the rock and the distance traveled by the boat. We can work this distance out because we know the boat sails with a uniform velocity of 96 meters per minute and the journey takes three minutes. If the boat covers 96 meters in each minute, it will cover 96 times three, that’s 288 meters, in three minutes.

Now, we’re looking to find the height of the rock. That’s this length here. And it isn’t a side in our triangle. However, it is a side length in the right triangle formed by the horizontal, the vertical, and the second line of sight. And we know one angle of 44 degrees in this triangle. This side formed by the line of sight is shared with the first triangle. So our approach is going to be to use the first triangle to try and work out this shared side and then use this shared side and the angle of 44 degrees in the right triangle to calculate the height of the rock.

Let’s look at this non-right triangle more closely then. We have an angle of 39 degrees, and we can work out each of the other two angles. For example, the obtuse angle is on a straight line with the angle of elevation of 44 degrees. So we can calculate it by subtracting 44 from 180, which gives 136 degrees. To calculate the final angle, we can use the angle sum in a triangle. 180 minus the other two angles of 136 and 39 gives five. So we have all three angles in this triangle.

If we want to work out a side length in a non-right triangle in which we know all three angles and one other side length, we can apply the law of sines, which tells us that the ratio between a side and the sine of its opposite angle is constant throughout the triangle. 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, which is equal to 𝑐 over sin 𝐶, where the lowercase letters represent sides and the uppercase letters represent angles.

Our side, the side highlighted in pink, which we can call 𝑥 meters, is opposite the angle of 39 degrees. And then we also know a side of 288 meters, which is opposite the angle of five degrees. So applying the law of sines, we have 𝑥 over sin of 39 degrees equals 288 over sin of five degrees. To solve this equation for 𝑥, we can multiply both sides by sin of 39 degrees, giving 𝑥 equals 288 sin 39 degrees over sin five degrees. Using a calculator, that evaluates to 2079.544. But we’ll keep the value as accurate as possible for now.

Now that we know the length of the shared side, we can consider the right triangle which contains the height we’re looking to calculate, which we can now call 𝑦 meters. Labeling the sides of this triangle in relation to the angle of 44 degrees, we want to calculate the opposite. And we know the hypotenuse. So recalling SOH CAH TOA, it is the sine ratio that we want to use. Remember, sine is equal to opposite over hypotenuse. So substituting, we have sin of 44 degrees is equal to the opposite 𝑦 over the hypotenuse 𝑥, which we’ve calculated to be 2079.544.

We can then multiply through by this value of 2079.544 to give 𝑦 equals 2079.544 multiplied by sin of 44 degrees. Now, of course, it makes sense to have kept that value of 2079.544 and so on our calculator display and then just type “multiplied by sin of 44 degrees” to give an answer that is as accurate as possible. Evaluating this gives 1444.57 continuing. We were asked to give our answer to the nearest meter, so the five in the first decimal place means we’re going to be rounding up, giving a height of 1445 meters to the nearest meter.

In this problem then, we used the law of sines in a non-right triangle and right-angle trigonometry in a right triangle in order to calculate this missing length. A key skill at the start of the question was drawing a diagram to represent the problem.

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