A lighthouse with a height of 89 meters is constructed on a cliff. From a boat in the sea, the angle of elevation to the top of the lighthouse is 72 degrees, and the angle of elevation to the base of the lighthouse is 34 degrees. Find the height of the cliff from sea level, giving the answer to the nearest meter.
Let’s begin by drawing a diagram to represent this situation. We have a lighthouse 89 meters tall on top of a cliff. There’s a boat in the sea and we’re told the angles of elevation to the top and base of the lighthouse. An angle of elevation is an angle measured from the horizontal when you look up towards an object. The angle of elevation from the boat to the top of the lighthouse is 72 degrees. The angle of elevation to the base of the lighthouse is 34 degrees. We’re asked to find the height of the cliff from sea level, which we’ll refer to as 𝑥 meters.
Now if we consider this diagram, we see that we have some right-angled triangles formed by the horizontal from the boat to the base of the cliff, the vertical line formed by the cliff and the lighthouse, and the line of sight from the boat to either the base or the top of the lighthouse. The two triangles have one side in common — the horizontal between the base of the cliff and the boat.
Our strategy is going to be to try to write down expressions for this common side using the two triangles. We’ll then see if we can equate them and solve in order to find the value of 𝑥. We’ll give the length of this shared side a value — 𝑦 meters — and consider the smaller triangle first of all. Remember this is the triangle with the angle of 34 degrees. In relation to this angle of 34 degrees, the sides 𝑥 and 𝑦 are in fact the opposite and adjacent sides of the right-angled triangle.
Recalling the acronym SOHCAHTOA, we remember that the ratio between these two sides is the tan ratio. The definition of tan is the opposite side divided by the adjacent. So we have that tan of 34 degrees is equal to 𝑥 over 𝑦. Multiplying both sides of this equation by 𝑦 gives 𝑦 tan 34 degrees is equal to 𝑥. Dividing both sides of the equation by tan of 34 degrees, which is just a number, gives our first expression for 𝑦. It’s equal to 𝑥 over tan of 34 degrees. So we have our first expression for 𝑦.
Next, we consider the larger triangle — the one with the angle of 72 degrees. In relation to this angle, 𝑦 is still the adjacent side. The opposite this time is the total height of the cliff and the lighthouse: 𝑥 plus 89. So the tan ratio in this triangle opposite divided by adjacent is tan of 72 degrees is equal to 𝑥 plus 89 over 𝑦.
Now we will rearrange this equation to give an expression for 𝑦. Multiplying both sides of the equation by 𝑦 gives 𝑦 tan 72 degrees is equal to 𝑥 plus 89. Dividing both sides of the equation by tan of 72 degrees gives our second expression for 𝑦: 𝑦 is equal to 𝑥 plus 89 over tan of 72 degrees.
Remember the purpose of writing down these two expressions for 𝑦 was so that we could then equate them, giving an equation in terms of 𝑥 only. Equating the two expressions gives 𝑥 over tan of 34 degrees is equal to 𝑥 plus 89 over tan of 72 degrees.
Now, I’m going to delete some of the earlier working out so that I’ve got a space to solve this equation. So, if you need to drop down any of the earlier working out, pause the video and do so now. So, the equation that we’d found was 𝑥 over tan 34 degrees is equal to 𝑥 plus 89 over tan of 72 degrees. The first step to solving this equation is to eliminate the denominators. So I’m going to cross multiply. This gives 𝑥 multiplied by tan of 72 degrees is equal to 𝑥 plus 89 multiplied by tan of 34 degrees.
Next, I’ll expand the bracket on the right-hand side giving 𝑥 tan 34 degrees plus 89 tan 34 degrees. Now, I have 𝑥s on both sides of the equation and I’d like to group them together on the left. So my next step is going to be to subtract 𝑥 tan 34 degrees from both sides. I now have 𝑥 tan 72 degrees minus 𝑥 tan 34 degrees is equal to 89 tan 34 degrees. The left-hand side of this equation can be factorized by taking a common factor of 𝑥 from both terms. This gives 𝑥 multiplied by tan 72 degrees minus tan 34 degrees is equal to 89 tan 34 degrees.
To find the value of 𝑥, I just need to divide both sides by the bracket. This gives me my expression for 𝑥. It’s equal to 89 tan 34 degrees over tan 72 degrees minus tan 34 degrees.
Now at this point, we can use our calculators to evaluate this, making sure that they’re in degree mode. Evaluating with a calculator should give 24.97997. If you haven’t got this value, check firstly either your calculator is indeed in degree mode and secondly check that you’ve used brackets correctly when typing in the fraction.
Looking back at the question, we’ve been asked to give our answer to the nearest meter. The height of the cliff from sea level to the nearest meter is 25 meters.