E and A are two hot air balloons flying at height 117 metres and 84 metres, respectively. The angles of depression at a point C on the ground from E and A are 40 degrees and 29 degrees, respectively. Find the distance between the two balloons giving the answer correct to the nearest metre.
Let’s start by identifying what we have and what we need. We have two right-angled triangles with a side and an angle each, which means we can calculate additional sides if necessary. We also have a non-right-angled triangle, one of the sides of which we are trying to find the length of. Ultimately, we’re going to be using either the sine rule or the cosine rule to find the value of this side labelled 𝑥. In order to do this, however, we do need to find some additional information.
Firstly, we can see that the line 𝐷𝐵 representing the ground is a straight line. We know the angles on a straight line add up to 180 degrees. So if we subtract 40 and 29 from 180, we find the value of the angle we labelled as 𝜃 to be 111 degrees. We still, however, need much more information to be able to find the value of 𝑥.
Let’s sketch separately this right-angled triangle from our diagram to help us see what to do next. It’s a right-angled triangle with one angle and one side, which means we can use trigonometry to find any additional sides. This triangle shares its hypotenuse with the larger non-right-angled triangle that we are ultimately interested in. We can therefore use the sine ratio to find its length.
Naming the hypotenuse 𝑎 and substituting our values in gives us sine 40 equals 117 over 𝑎. Multiplying both sides by 𝑎 give us 𝑎 sine 40 equals 117. And then to solve, we divide through by sine 40. 𝑎 is equal to 182.01 metres. You’ll notice we haven’t rounded our answer yet. Instead we’ll need to use this exact value in our further calculations.
Now, let’s repeat this process with the remaining right-angled triangle. Once again, the triangle shares its hypotenuse with the larger non-right-angled triangle. Let’s call this 𝑏 for clarity. Using the sine ratio to find the length of the hypotenuse, we get sine 29 is 84 over 𝑏 Rearranging gives us 𝑏 equals 84 over sin 29. 𝑏 is equal to 173.26. We now have a non-right-angled triangle with two given sides and one angle.
Redrawing the diagram to label the sides to match our formula, we can apply the cosine rule to help us evaluate the length of the side we labelled as 𝑥. Substituting these values in gives us 𝑥 squared equals 182.01 squared add 173.26 squared minus two times 182.01 times 173.26 times cos 111.
Remember when we type this into our calculator, we need to make sure we use those saved values for 𝑎 and 𝑏, rather than rounding too early. 𝑥 squared is therefore equal to 85620.14. And then to solve this equation, we need to square root both sides. 𝑥 is equal to 292.6.
The distance between the balloons is 293 metres correct to the nearest metre.