Question Video: Solving Word Problems Involving Angles of Elevation Mathematics

Video Transcript

A person measured the angle of elevation of a fixed hot-air balloon as 𝜋 by seven. He then walked 522 meters in a horizontal direction towards the balloon. And the angle of elevation was 𝜋 by four. Find the height of the hot-air balloon to the nearest meter.

We’ll begin by drawing a diagram. We’re told that the person initially measured the angle of elevation to the hot-air balloon as 𝜋 by seven. So we’re working in radians. An angle of elevation is the angle measured from the horizontal when we look up towards something. So, in this case, the person is looking up from the ground to a hot-air balloon in the sky. The person then walks 522 meters in a horizontal direction. So they’re walking along the ground towards the balloon. From this new position, the angle of elevation of the balloon is 𝜋 by four radians.

Now, we need to be careful when we mark this in our diagram. It’s the angle from the horizontal as we look up towards the object. So it’s this angle here. We’ve now drawn our diagram. And we can see that we have a triangle formed by the two lines of sight up to the balloon from each position and the horizontal distance of 522 meters. What we want to find though is the height of the hot-air balloon above the ground. So we’ll include a line to represent this on our diagram, and we can label this as 𝑥 meters. We’ll also add some letters to our diagram: 𝐵 for the position of the balloon, 𝐴 and 𝐶 for the two positions where this man is standing, and 𝐷 for the point vertically below the balloon on the ground.

Now, we can see that the height of the balloon, which we’ve labeled as 𝑥 meters, is one side in the right triangle 𝐵𝐶𝐷. But the only other information we know about this triangle is that it is a right triangle and it has another angle of 𝜋 by four radians. Without knowing any lengths in this triangle, we won’t be able to calculate the length of the side 𝐵𝐷. Notice though that the side 𝐵𝐶 is shared with the other triangle in the diagram, triangle 𝐴𝐵𝐶, in which we know there is an angle of 𝜋 by seven radians and a side of length of 522 meters.

We can also work out the measure of angle 𝐴𝐶𝐵 by recalling that angles on a straight line sum to 180 degrees or 𝜋 radians. So the measure of angle 𝐴𝐶𝐵 is 𝜋 minus 𝜋 by four, which is three 𝜋 by four radians. We can also work out the measure of the third angle in this triangle, angle 𝐴𝐵𝐶, by recalling that angles in any triangle sum to 180 degrees or 𝜋 radians. So the measure of this third angle is 𝜋 minus 𝜋 by seven minus three 𝜋 by four, which is three 𝜋 by 28 radians.

In triangle 𝐴𝐵𝐶, we now know the measures of all three angles and the length of one side. This means that we can calculate the length of any other side in triangle 𝐴𝐵𝐶 by applying the law of sines. This tells us that, in any triangle with angles capital 𝐴, capital 𝐵, and capital 𝐶 and opposite sides labeled with the corresponding lowercase letters 𝑎, 𝑏, and 𝑐, the ratio between the length of each side and the sine of its opposite angle is constant. 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵, and this is equal to 𝑐 over sin 𝐶.

In our triangle, the side of 522 meters is opposite the angle of three 𝜋 by 28 radians. And the side we want to calculate, side 𝐵𝐶, is opposite the angle of 𝜋 by seven radians. So, using the law of sines, we can form an equation. 𝐵𝐶 over sin of 𝜋 by seven equals 522 over sin of three 𝜋 by 28. Multiplying both sides of this equation by sin of 𝜋 by seven gives an expression for the length of 𝐵𝐶: 522 sin of 𝜋 by seven over sin of three 𝜋 by 28.

Now, we can evaluate this on our calculators, which we must ensure are in radian mode. And it gives 685.7452 continuing. Make sure you keep that exact value on your calculator display or store it in the calculator memory.

Looking at triangle 𝐵𝐶𝐷, we now know the length of one side, side 𝐵𝐶. We know that it’s a right triangle, and we know the measure of one other angle. Angle 𝐵𝐶𝐷 is 𝜋 by four radians. We can therefore apply right triangle trigonometry to calculate the length of the side 𝐵𝐷. We begin by labeling the three sides of the triangle in relation to the angle of 𝜋 by four radians. Side 𝐵𝐷 is the opposite, side 𝐶𝐷 is the adjacent, and side 𝐵𝐶 is the hypotenuse. The side we wish to calculate is the opposite, and the side whose length we know is the hypotenuse. So, recalling the acronym SOH CAH TOA, we can answer this question using the sine ratio. The sine ratio in a right triangle is the length of the opposite divided by the length of the hypotenuse. So we have that sin of 𝜋 by four is equal to 𝑥 over 685.7452 continuing. To solve for 𝑥, we need to multiply both sides of this equation by 685.7452 continuing, giving 𝑥 equals 685.7452 continuing times sin of 𝜋 by four.

Now, if you’ve kept that value on your calculator display, you can now just type multiplied by sin of 𝜋 by four to give the answer. And doing so gives 484.895 continuing. So we’ve found the height of the balloon. The question asked that we give our answer to the nearest meter. So we need to round to the nearest integer. That’s 485. So, by recalling the angles of elevation, our angles measured from the horizontal when we look up towards something, and then applying the law of sines and then the sine ratio in a right triangle, we found the height of this hot-air balloon to the nearest meter is 485 meters.