A flag is hung 22 meters up a flagpole. As the flag is raised, the angle of elevation from a point 21 meters away from the base of the flagpole to the flag is 74 degrees. Find the increase in height of the flag, giving the answer to two decimal places.
It would probably be helpful here to sketch this image. First of all, here’s our flag, and the flag is hung 22 meters up. As the flag is being raised, from there, we’re given an angle of elevation from a point 21 meters away from the base of the flagpole. This will be the line created by the angle of elevation.
Our angle here is 74 degrees. Given that angle of elevation, we can use a right triangle to solve this problem. We have an adjacent side length to our angle, and we’re missing the opposite side length to our angle. The ratio of opposite to adjacent is a tangent ratio. In our case, tangent of 74 degrees equals the height of the flagpole over 21 meters. And that means, by multiplying tangent of 74 degrees by 21, we’ll find the height of the flagpole.
When we calculate this value using some form of technology, we get a repeating decimal value, 73.2357. We’re interested in the height to the nearest two decimal places. We want to round to the hundredths place. Because the digit in the thousandths place is five or greater, we round up in the hundredths place. The three becomes a four, and everything to the left of the hundredths place stays the same. We now have a height of 73.24 meters.
We have to be careful here because the whole height of the flagpole is 73.24 meters and our question is only interested in the increase in height. Remember that our flag started 22 meters up. We need to take the total height, 73.24 meters, and subtract the flag’s starting height. 73.24 meters minus 22 meters equals 51.24 meters. Our flag was raised by an amount of 51.24 meters.