### Video Transcript

A person measured the angle of
elevation of a fixed hot-air balloon as π by seven. He then walked 522 meters in a
horizontal direction towards the balloon. And the angle of elevation was π
by four. Find the height of the hot-air
balloon to the nearest meter.

Weβll begin by drawing a
diagram. Weβre told that the person
initially measured the angle of elevation to the hot-air balloon as π by seven. So weβre working in radians. An angle of elevation is the angle
measured from the horizontal when we look up towards something. So, in this case, the person is
looking up from the ground to a hot-air balloon in the sky. The person then walks 522 meters in
a horizontal direction. So theyβre walking along the ground
towards the balloon. From this new position, the angle
of elevation of the balloon is π by four radians.

Now, we need to be careful when we
mark this in our diagram. Itβs the angle from the horizontal
as we look up towards the object. So itβs this angle here. Weβve now drawn our diagram. And we can see that we have a
triangle formed by the two lines of sight up to the balloon from each position and
the horizontal distance of 522 meters. What we want to find though is the
height of the hot-air balloon above the ground. So weβll include a line to
represent this on our diagram, and we can label this as π₯ meters. Weβll also add some letters to our
diagram: π΅ for the position of the balloon, π΄ and πΆ for the two positions where
this man is standing, and π· for the point vertically below the balloon on the
ground.

Now, we can see that the height of
the balloon, which weβve labeled as π₯ meters, is one side in the right triangle
π΅πΆπ·. But the only other information we
know about this triangle is that it is a right triangle and it has another angle of
π by four radians. Without knowing any lengths in this
triangle, we wonβt be able to calculate the length of the side π΅π·. Notice though that the side π΅πΆ is
shared with the other triangle in the diagram, triangle π΄π΅πΆ, in which we know
there is an angle of π by seven radians and a side of length of 522 meters.

We can also work out the measure of
angle π΄πΆπ΅ by recalling that angles on a straight line sum to 180 degrees or π
radians. So the measure of angle π΄πΆπ΅ is
π minus π by four, which is three π by four radians. We can also work out the measure of
the third angle in this triangle, angle π΄π΅πΆ, by recalling that angles in any
triangle sum to 180 degrees or π radians. So the measure of this third angle
is π minus π by seven minus three π by four, which is three π by 28 radians.

In triangle π΄π΅πΆ, we now know the
measures of all three angles and the length of one side. This means that we can calculate
the length of any other side in triangle π΄π΅πΆ by applying the law of sines. This tells us that, in any triangle
with angles capital π΄, capital π΅, and capital πΆ and opposite sides labeled with
the corresponding lowercase letters π, π, and π, the ratio between the length of
each side and the sine of its opposite angle is constant. π over sin π΄ equals π over sin
π΅, and this is equal to π over sin πΆ.

In our triangle, the side of 522
meters is opposite the angle of three π by 28 radians. And the side we want to calculate,
side π΅πΆ, is opposite the angle of π by seven radians. So, using the law of sines, we can
form an equation. π΅πΆ over sin of π by seven equals
522 over sin of three π by 28. Multiplying both sides of this
equation by sin of π by seven gives an expression for the length of π΅πΆ: 522 sin
of π by seven over sin of three π by 28.

Now, we can evaluate this on our
calculators, which we must ensure are in radian mode. And it gives 685.7452
continuing. Make sure you keep that exact value
on your calculator display or store it in the calculator memory.

Looking at triangle π΅πΆπ·, we now
know the length of one side, side π΅πΆ. We know that itβs a right triangle,
and we know the measure of one other angle. Angle π΅πΆπ· is π by four
radians. We can therefore apply right
triangle trigonometry to calculate the length of the side π΅π·. We begin by labeling the three
sides of the triangle in relation to the angle of π by four radians. Side π΅π· is the opposite, side
πΆπ· is the adjacent, and side π΅πΆ is the hypotenuse. The side we wish to calculate is
the opposite, and the side whose length we know is the hypotenuse. So, recalling the acronym SOH CAH
TOA, we can answer this question using the sine ratio. The sine ratio in a right triangle
is the length of the opposite divided by the length of the hypotenuse. So we have that sin of π by four
is equal to π₯ over 685.7452 continuing. To solve for π₯, we need to
multiply both sides of this equation by 685.7452 continuing, giving π₯ equals
685.7452 continuing times sin of π by four.

Now, if youβve kept that value on
your calculator display, you can now just type multiplied by sin of π by four to
give the answer. And doing so gives 484.895
continuing. So weβve found the height of the
balloon. The question asked that we give our
answer to the nearest meter. So we need to round to the nearest
integer. Thatβs 485. So, by recalling the angles of
elevation, our angles measured from the horizontal when we look up towards
something, and then applying the law of sines and then the sine ratio in a right
triangle, we found the height of this hot-air balloon to the nearest meter is 485
meters.