Video Transcript
Two points on the ground lie
collinearly on the same side of a flagpole 29 meters tall. The angles of elevation from the
two points to the top of the flagpole are 45 degrees 18 minutes and 34 degrees 18
minutes. Find the distance between the two
points giving the answer to one decimal place.
Let’s begin by sketching this
situation. There’s a flagpole which is 29
meters tall. There are then two points on the
ground that lie collinearly on the same side of the flagpole. This just means that they lie on
the same straight line. The angles of elevation from the
two points to the top of the flagpole are 45 degrees 18 minutes and 34 degrees 18
minutes. Now, angles of elevation are angles
that are measured from the horizontal to the line of sight when we look up towards
something. So in this case, we’re looking up
from the ground to the top of the flagpole.
We may also want to convert these
angles from angles that are measured in degrees and minutes to angles that are
measured solely in degrees. We can recall that there are 60
minutes in a degree, so 18 minutes is eighteen sixtieths of a degree. The angle 45 degrees and 18 minutes
is therefore 45 and eighteen sixtieths degrees, which is 45.3 degrees. For the same reason, the angle 34
degrees 18 minutes is 34.3 degrees.
So we can add the measures of these
two angles to our diagram. And we do need to be careful that
in each case, we are starting at the horizontal and then looking up towards the
flagpole when we measure the angle.
We want to find the distance
between the two points, which we’ve now labeled as 𝐴 and 𝐵, and we’re gonna call
this distance 𝑦 meters. 𝑦 is the difference between 𝐴𝐶,
that’s the distance between the bottom of the flagpole and the point furthest away,
and 𝐵𝐶, the distance between the bottom of the flagpole and the point closest to
it.
We can work each of these distances
out using right triangle trigonometry. Consider first the right triangle
𝐵𝐶𝐷, in which we know the length of the side 𝐶𝐷 and the measure of the angle
𝐶𝐵𝐷. In relation to the angle of 45.3
degrees, the side of 29 meters is the opposite, 𝐵𝐶 is the adjacent, and 𝐵𝐷 is
the hypotenuse. We know the length of the opposite
side, and we want to calculate the length of the adjacent side. So recalling the acronym SOH CAH
TOA, it is the tangent ratio that we need to use.
For an angle 𝜃 in a right
triangle, the tan of angle 𝜃 is defined to be equal to the length of the opposite
side divided by the length of the adjacent side. So we have tan of 45.3 degrees is
equal to 29 over 𝐵𝐶. Multiplying both sides of this
equation by the unknown 𝐵𝐶 gives 𝐵𝐶 times tan of 45.3 degrees equals 29. To find 𝐵𝐶, we divide both sides
of the equation by tan of 45.3 degrees, giving 𝐵𝐶 equals 29 over tan 45.3
degrees. Evaluating this on a calculator,
which must be in degree mode, we find that the length of 𝐵𝐶 is 28.697 continuing
meters.
So we know the length of 𝐵𝐶. That’s the distance between the
bottom of the flagpole and the nearest point. And we now need to calculate the
length of 𝐴𝐶, the distance between the bottom of the flagpole and the furthest
away point. To do this, we consider the right
triangle 𝐴𝐶𝐷.
In this triangle, in relation to
the known angle of 34.3 degrees, the side 𝐶𝐷 is the opposite, the side 𝐴𝐶 is the
adjacent, and the side 𝐴𝐷 is the hypotenuse. As before, the side whose length we
know is the opposite, and the side we wish to calculate is the adjacent. So we’re using the tan ratio. We have that tan of 34.3 degrees is
equal to 29 over 𝐴𝐶. The method for calculating 𝐴𝐶 now
proceeds in exactly the same way as it did when we were calculating 𝐵𝐶. First, we multiply by 𝐴𝐶. Then, we divide by tan of 34.3
degrees to give 𝐴𝐶 equals 29 over tan of 34.3 degrees, which is 42.512 continuing
meters.
We now know the distances between
the bottom of the flagpole and each point. So all that remains is to find the
distance between the two points by subtracting 𝐵𝐶 from 𝐴𝐶. We have 42.512 continuing minus
28.697 continuing, which gives 13.81 continuing. The question specifies that we
should give our answer to one decimal place. So we round to 13.8, and we found
that the distance between the two points to one decimal place is 13.8 meters.
It’s worth pointing out that there
is a different method we could’ve used to answer this question. Looking at triangle 𝐵𝐶𝐷 first of
all, we could’ve used the sine ratio to calculate the length of side 𝐵𝐷, which is
the hypotenuse. As this side is shared with
triangle 𝐴𝐵𝐷, we now know one side length in this triangle. We could also calculate the measure
of angle 𝐴𝐵𝐷 by recalling that angles on a straight line sum to 180 degrees. So the measure of this angle is 180
degrees minus 45.3 degrees.
We could then calculate the measure
of the third angle in this triangle, angle 𝐵𝐷𝐴, by recalling that angles in a
triangle sum to 180 degrees. So we could subtract the measures
of the other two angles from 180 degrees. As we now know the measures of all
three angles in this triangle and the length of one side, we could apply the law of
sines to calculate the length 𝐴𝐵. If we were to apply this method, we
would of course get the same result, which is that the distance between the two
points to one decimal place is 13.8 meters.
