The height of a lighthouse is 60 meters. The angles of elevation between two boats in the sea and the top of the lighthouse are 29 degrees and 39 degrees, respectively. Given that the two boats and the base of the lighthouse are colinear and that the boats are both on the same side of the lighthouse, find the distance between the two boats giving the answer to the nearest meter.
So, to help us understand what’s going on, I’m gonna draw a sketch. First of all, I’ve drawn the lighthouse which is 60 meters tall. And then, we have two boats. And these two boats are colinear with the base of the lighthouse. So, they’re gonna be in a line with the base of the lighthouse. And also, they’re on the same side of the lighthouse. So, I’ve drawn them on the same side of the lighthouse and I’ve called them 𝐴 and 𝐵.
And then, we’re also told that the angles of elevation between the two boats and the top of the lighthouse are 29 degrees and 39 degrees, respectively. And the angle of elevation is the angle made from the horizontal then up to the point, which we’re looking for, which is the top of the lighthouse. So, what we’ve created here are two right-angled triangles. And we can use them to try and solve the problem. And to solve the problem, what we need to do is find the distance between our boats 𝐴 and 𝐵.
So, to find this distance, first what we need to do is find the distance from boat 𝐴 to the lighthouse and the distance from boat 𝐵 to the lighthouse. So, we’re gonna start with boat 𝐴 to the lighthouse. And what I’ve done is I’ve drawn a right-angled triangle here in blue to help us see what we’re working with. As we’re looking at a right-angled triangle, we could either use the Pythagorean theorem or trig ratios. Well, we’re gonna use trig ratios because we’ve got an angle and a side.
So, now what I do is I label the sides. We’ve got the hypotenuse which is opposite the right angle and the longest side. We’ve got the opposite which is opposite our angle of 39 degrees. And we have the adjacent. And the adjacent is next to the angle and between the angle and the right angle.
Now, what we need to do is decide which ratio to use. Well, we have the opposite and we want to find the adjacent. So, we’ve got O and A. So therefore, we’re gonna use the TOA part of SOH CAH TOA, our memory aid. And what this tells us is that the tan of 𝜃, so the tangent of 𝜃 our angle, is gonna be equal to the opposite divided by the adjacent.
So therefore, if we substitute in our values, we’re gonna have tan of 39 is gonna be equal to 60, cause that’s our opposite, divided by 𝑥, because 𝑥 is our adjacent. So then, if we multiply each side of the equation by 𝑥, we’re gonna get 𝑥 tan 39 is equal to 60. And then, as we wanna find 𝑥, what we do is divide each side of the equation by tan 39. So therefore, 𝑥 is gonna be equal to 60 over tan 39.
And just to maintain accuracy, I’m not gonna calculate this yet. I’ll only calculate it when I look to find the final answer. And that’s the distance between 𝐴 and 𝐵. Okay, so, now we know what 𝑥 is, so the distance from the lighthouse to 𝐴. We need to find out what the distance from the lighthouse to 𝐵 is going to be.
So, now like I said, what we want to do is find out the distance from the lighthouse, so the base of the lighthouse, to 𝐵. And what I’ve called that in our diagram is 𝑦. This time, we’re gonna have a right-angled triangle, which I have drawn in green. But the angle we’re interested in this time is going to be 29 degrees. So, once again, because we’ve got the opposite and the adjacent, because the opposite is 60 meters again, but the adjacent is what we’re looking, so that’s our 𝑦, then we’re gonna use the tan ratio once more.
So, we’re gonna get tan 29 is equal to 60 over 𝑦. And then, if we rearrange in the same way as previously, we’re gonna get 𝑦 is equal to 60 over tan 29. And that’s cause we multiplied by 𝑦 and then divide by tan 29. So therefore, the distance from 𝐴 to 𝐵 is going to be equal to 𝑦 minus 𝑥. So therefore, this is gonna be equal to 60 over tan 29 minus 60 over tan 39. And this is gonna give us an answer of 34.149, et cetra.
So, now what we’ve got to do decide how to leave our answer. Well, if we check the question, the question wants us to leave it to the nearest meter. So therefore, if we round 34.149 to the nearest meter, we’re gonna get 34 meters. And that’s because the digit after the four was a one, so we keep the four the same. So therefore, we can say that the distance between the two boats is 34 meters.