### Video Transcript

A building is eight meters
tall. The angle of elevation from the top
of the building to the top of the tree is 44 degrees and the angle of depression
from the top of the building to the base of the tree is 58 degrees. Find the distance between the base
of the building and the base of the tree giving the answer to two decimal
places.

So let’s begin by sketching this
problem. First, we have a building which is
eight meters tall. We’re then told about an angle of
elevation from the top of this building to the top of a tree. Remember, an angle of elevation is
measured from the horizontal up towards something. So this tree is taller than the
building. So we add the tree and the angle of
elevation of 44 degrees. Next, we’re told that the angle of
depression from the top of the building to the base of the tree is 58 degrees. We need to be careful when labeling
this angle. Remember that an angle of
depression is measured from the horizontal down towards something, so the angle of
58 degrees is this angle here.

What we’re looking to calculate is
the distance between the base of the building and the base of the tree, which is
this length here. Now, looking carefully at our
diagram, we can see that we have a right triangle. And the length we’re looking to
calculate, which we’ll call 𝑦 meters, is one of the sides. We also know another of the
sides. It’s the height of the building,
eight meters. We can also work out one of the
angles inside our triangle. If we subtract the angle of
depression of 58 degrees from a right angle of 90 degrees, we can find the top angle
in our triangle. It’s 32 degrees. So we now have a right triangle in
which we know one length and one other angle, and we wish to calculate a second
length.

Labeling the sides of the triangle
in relation to the 32-degree angle, we know the adjacent and we want to calculate
the opposite. So we’re going to use the tan
ratio. Recall, the tan ratio is opposite
over adjacent, so we have tan of 32 degrees is equal to 𝑦 over eight. Multiplying both sides of this
equation by eight, we have that 𝑦 is equal to eight tan 32 degrees. Evaluating this on a calculator,
which must be in degree mode, we have 4.9989.

We’re asked to give our answer to
two decimal places. So starting with the eight in the
third decimal place and rounding up, we have 5.00 meters. So we’ve completed the problem. The distance between the base of
the building and the base of the tree to two decimal places is 5.00 meters.