A tower is 33 meters tall. The angle of depression from the top of a hill to the top of the tower is 31 degrees. The angle of depression from the top of the hill to the bottom of the tower is 52 degrees. Find the height of the hill given the bases of the hill and the tower lie on the same horizontal level. Give the answer to the nearest meter.
Let’s begin this problem by sketching a diagram of the situation. Firstly, we’re told that we have a tower which is 33 meters tall. We also have a hill whose height we don’t know, and we’re given various information about some angles of depression. Remember that an angle of depression is the angle formed between the horizontal and your eye line or the line of sight as you look down towards an object. We’re told that the angle of depression from the top of the hill to the top of the tower is 31 degrees. So that’s this angle here, the angle between the horizontal and the line of sight as we look down at the top of the tower from the top of the hill.
We’re also told that the angle of depression from the top of the hill to the bottom of the tower is 52 degrees. So that would be this angle here, the angle between the horizontal and the line of sight as we look down towards the bottom of the tower. We can therefore divide the angle of 52 degrees up into a 31-degree angle, which we’ve already sketched, and the remaining 21 degrees, which is between the two lines of sight. Now, remember, this diagram is not to scale.
We’re asked to find the height of the hill, so we can call this 𝑥 meters. And the final piece of information is that the base of the hill and the base of the tower lie on the same horizontal plane, which just means that we can assume the ground between them is flat. And so there is a right angle between the ground and the vertical height of the hill or, indeed, the vertical height of the tower.
We can now see that there are two triangles in our diagram: this triangle here, a nonright triangle in which we know a length of 33 meters and an angle of 21 degrees, and this triangle here, a right triangle in which we currently don’t know any other information. These two triangles have a shared side, which we can call 𝑦 meters. Now as we don’t currently know any further information in our orange triangle, let’s consider instead the pink triangle. If we sketch in another horizontal here, then we can see that we have a pair of alternate angles in parallel lines. And we know that alternate angles are equal, so this portion of the angle inside our triangle is also 31 degrees.
We also know that the angle between the horizontal and vertical is 90 degrees. And so this obtuse angle in the pink triangle is composed of a 90-degree angle and a 31-degree angle. It’s therefore equal to 121 degrees. We can also work out the third angle in our triangle because we know that the angle sum in any triangle is 180 degrees. So the final angle is 180 degrees minus the other two angles, which are 121 degrees and 21 degrees, meaning that the third angle is 38 degrees. If we consider the pink triangle then, we now know the measures of all three angles and we know the length of one side, which means we have enough information to be able to calculate the length of either of the other two sides or, in particular, the side of length 𝑦 meters by applying the law of sines.
The law of sines tells us that in any triangle, the ratio between the length of a side and the sine of its opposite angle is constant, which we can write as 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶, where the lowercase letters represent sides and the uppercase letters represent their opposite angles. The side of 33 meters is opposite the angle of 21 degrees, and the side we wish to calculate, 𝑦, is opposite the angle of 121 degrees. So substituting into the law of sines, we have 𝑦 over sin of 121 degrees equals 33 over sin of 21 degrees. We can solve this equation for 𝑦 by multiplying both sides by sin of 121 degrees, giving 𝑦 equals 33 sin 121 degrees over sin of 21 degrees.
Evaluating on a calculator, we have that 𝑦 is equal to 78.931. So we now know the length of the shared side 𝑦, which means we have some more information about our orange triangle. We can also work out the measures of one of the other angles in this triangle. In the bottom left of our diagram, we have a right angle between the horizontal and vertical. And we know that part of this angle is 38 degrees. The remaining portion of this angle then will be 90 degrees minus 38 degrees. That’s 52 degrees. So in our orange triangle, which is a right triangle, we now know one angle of 52 degrees, we know one side of 78.9 meters, and we wish to calculate the length of another side, which means we can apply right-angle trigonometry.
In relation to the angle of 52 degrees, the side we wish to calculate is the opposite. And the side we know of 78.9 meters is the hypotenuse. The trigonometric ratio we need to use then which involves the opposite and hypotenuse is the sine ratio. This is just the standard sine ratio in a right triangle, not the law of sines. Remember, sin is opposite over hypotenuse, so we have sin of 52 degrees is equal to 𝑥 over 78.9 continuing. We can find the value of 𝑥 by multiplying both sides of this equation by 78.9. So we have 𝑥 equals 78.9 multiplied by sin of 52 degrees. Now we should’ve kept that exact value on our calculated displays, which means we can now just type “multiplied by sin of 52 degrees” and be sure that there are no rounding errors in our answer.
When we evaluate this, making sure our calculator is in degree mode, we get 62.198 continuing. The question asked us to give our answer to the nearest meter. So as the first decimal place is a one, we round down to 62 meters. So by applying the law of sines and then the sine ratio in a right triangle, we found that the height of the hill to the nearest meter is 62 meters.