Video Transcript
A palm tree 10.6 meters tall is observed from a point 12 meters away on the same horizontal plane as the base of the tree. Find the angle of elevation to the top of the palm tree, giving the answer to the nearest minute.
We’ll begin by sketching this problem. There is a palm tree which is 10.6 meters tall. It is observed from a point 12 meters away on the same horizontal plane as the base of the tree, which means that the angle between the two lines we’ve drawn is a right angle. We’re asked to calculate the angle of elevation to the top of the palm tree.
Now, an angle of elevation is the angle formed between the horizontal and the line of sight when we look up towards an object. This is in contrast to an angle of depression, which is the angle formed between the horizontal and the line of sight when we look down towards an object. In our diagram, the horizontal is the side of length 12 meters. And if we then look up towards the palm tree, the angle of elevation is this angle here. Let’s call this angle 𝜃.
Now we can see that we have a right triangle in which we know the lengths of two sides and we want to calculate the measure of one of the angles. We can do this using right triangle trigonometry. We’ll begin by labeling the sides of the triangle in relation to this angle 𝜃. The side of length 10.6 meters is the opposite side. And the side of length 12 meters is the adjacent. Recalling the acronym SOH CAH TOA, it is the tangent ratio that we need to use in this question. That’s defined as follows. For an angle 𝜃 in a right triangle, the tan of angle 𝜃 is equal to the length of the opposite side divided by the length of the adjacent side.
We know these two values for this triangle, and so we can form an equation. tan of 𝜃 is equal to 10.6 over 12. To solve this equation for 𝜃, we need to apply the inverse tangent function. So we have 𝜃 is equal to the inverse tan of 10.6 over 12. We can now evaluate this on a calculator, ensuring that our calculator is in degree mode. And it gives 41.45523 continuing.
Now the question specifies that we should in fact give our answer to the nearest minute. For our value of 𝜃, we have 41 degrees and then this decimal of 0.45523 continuing, which we need to convert to minutes. Recalling that there are 60 minutes in one degree, we can work out the number of minutes represented by this decimal by multiplying it by 60. This gives 27.314 continuing, which to the nearest integer is 27. To the nearest minute then, this value of 41.45523 continuing degrees is 41 degrees and 27 minutes.
And so by recalling that an angle of elevation is the angle measured between the horizontal and the line of sight when we look up towards an object and then applying right triangle trigonometry, we found that the angle of elevation to the top of the palm tree to the nearest minute is 41 degrees and 27 minutes.
