Video Transcript
The distance between two buildings
is 40 meters. The top of building 𝐶𝐷 has an
angle of elevation of 30 degrees, measured from the top of building 𝐴𝐵. If the height of building 𝐴𝐵
equals 30 meters and the bases of the two buildings are on the same horizontal
plane, then the height of 𝐶𝐷 to the nearest meter equals how many meters.
So we’ve been given the diagram to
go alongside this problem. And all the information in the
question has been labeled on it. We have an angle of elevation here
of 30 degrees formed between the horizontal and the line of sight as we look up from
building 𝐴𝐵 to building 𝐶𝐷. We also know the height of building
𝐴𝐵, it’s 30 meters, and the horizontal distance between the two buildings, 40
meters. We’re told that the two buildings
are on the same horizontal plane, which simply means we can assume that the ground
between them is flat.
What we’re looking to calculate is
the height of building 𝐶𝐷. From the diagram, we can see that
this will be composed of two lengths, a portion which is the same height as building
𝐴𝐵, so that’s 30 meters, and a portion which is this currently unknown length
here, which we can think of as 𝑥 meters. We can also see that this length 𝑥
is one side in a right triangle. And in this triangle, we know one
other side of 40 meters and one angle of 30 degrees. We can therefore apply right-angle
trigonometry to calculate 𝑥.
Labeling the sides of this triangle
in relation to the 30-degree angle, we can see that we know the adjacent and we want
to calculate the opposite. So it’s the tan ratio that we’re
going to use. Recalling that tan is opposite over
adjacent, we have that tan of 30 degrees is equal to 𝑥 over 40. Multiplying both sides of this
equation by 40, we have that 𝑥 is equal to 40 multiplied by tan of 30 degrees. And evaluating this on a
calculator, ensuring our calculator is in degree mode, we find that 𝑥 is equal to
23.0940 continuing.
We haven’t quite finished though
because we need the total height of building 𝐶𝐷. So we need to add on the additional
length of 30 meters. Doing so gives 53.0940. We’re asked to give our answer to
the nearest meter. So as the digit in the first
decimal place is a zero, we round to 53. So by applying trigonometry in the
right triangle formed by the horizontal, the vertical, and the line of sight, we
found that the height of building 𝐶𝐷 to the nearest meter is 53 meter.
