Video Transcript
Anthony and Victoria want to find the height of a statue. Anthony stands five meters from the base of the statue and measures the angle of elevation, from the ground, to be 65 degrees. Victoria stands directly behind Anthony. She measures the angle of elevation, from the ground, to be 30 degrees. They both calculate the same height for the statue. How far behind Anthony must Victoria be standing? Give your solution to two decimal places.
We know that there is a statue and that Anthony is standing five meters away from the base. He measures the angle of elevation to the top of the statue to be 65 degrees. Victoria is standing some distance behind Anthony. We’ll call it 𝑑. She also measures the angle of elevation from the ground and she gets 30 degrees. We know that they both get the same measure for the height, but we’re not told what that value is, so we’ll call it ℎ. We can say that there is a right angle between the ground and the statue, and that means we can use our trig ratios to solve for the missing height.
If we start at 65 degrees, we have an adjacent side, five meters, and the opposite side ℎ, which represents the height of the statue. And when we’re dealing with the opposite and adjacent side lengths, that is the tangent of an angle. And so, we say that the tangent of 65 degrees equals the opposite side length, which for us will be the height of the statue, over the adjacent side length, which is five meters. To find the height, we multiply both sides of the equation by five meters. The height of the statue will equal five times the tangent of 65 degrees. What we get is 10.72253 continuing meters.
If you didn’t get this value, you should check your calculator and make sure it’s set to degrees and not to radians. Now, this is not actually the final solution because what we’re trying to find out is 𝑑, the distance Victoria is behind Anthony. But we will need to know the height of the statue to find the value of 𝑑. Again, we have a right angle between the ground and the statue, and again we have a tangent ratio.
If we began at 30 degrees, the opposite side length will be ℎ, the height of the statue. But we need to be careful about what the adjacent side length is. This variable, 𝑑, represents how far Victoria is from Anthony, but Anthony is already five meters away from the statue. And that means the adjacent side here is five plus 𝑑 meters away.
Tangent of 30 degrees is equal to ℎ over five plus 𝑑. We can’t solve this problem because we have two variables, but we can plug in what we already know the height to be, 10.72253 continuing. And now, we need to use this information to solve for 𝑑. Because five plus 𝑑 is in the denominator, we can get it out by multiplying both sides of the equation by five plus 𝑑. Which will give us five plus 𝑑 times tangent of 30 degrees is equal to 10.72253 continuing. From there, we divide both sides by tangent of 30 degrees. 10.72253 continuing divided by tangent of 30 degrees is equal to 18.57197 continuing.
Again, if you didn’t get this answer, you need to make sure that you’re operating in degrees and not in radians. From there, we subtract five from both sides to get 𝑑 equals 13.57197 continuing. We’re rounding to two decimal places. There is a seven in the hundreds place, then we need to look to the right of that seven. There is a one, which means we’ll round down to 13.57. This is a measure of meters because that’s the units we began with. This means that Victoria was about 18.57 meters away from the statue. But the question asked how far away from Anthony she was, which is 13.57 meters.
