Video Transcript
Two aircraft scramble from the same
airfield, one climbing at an angle of 45 degrees from the ground and the other at 20
degrees from the ground, as shown in the diagram. When both aircraft are horizontally
2500 meters from the airfield, the aircraft that climbed at a less steep angle is
vertically ℎ one meters above the ground and the aircraft that climbed more steeply
is vertically ℎ two meters above the ground. How far vertically is ℎ two above ℎ
one, to the nearest meter?
Okay, so we’re told in this
question that we’ve got two aircraft, each climbing at a different angle relative to
the ground as it leaves an airfield. This is shown in the diagram, where
we can see that we’ve got one aircraft at an angle of 45 degrees to the ground and
the other at an angle of 20 degrees. Also shown in the diagram is that
when both aircraft have traveled a distance of 2500 meters horizontally, the
steeper-climbing aircraft has reached a height of ℎ two meters and the less steep
one has reached a height of ℎ one meters. So all of the information given to
us in the main bit of question text is also shown in the diagram. This means we can safely get rid of
this text to clear ourselves some space.
Now, we’re asked to find the
difference in height between the two aircraft after both aircraft are horizontally
2500 meters from the airfield. That’s this distance here between
the height ℎ one and the height ℎ two in our diagram. In other words, the question is
asking us to calculate the value of ℎ two minus ℎ one. This means we need to start by
calculating the values of ℎ two and ℎ one.
We can notice by the way that the
blue lines showing the paths of the aircraft are effectively the displacement
vectors for the two aircraft. So the quantities ℎ two and ℎ one
that we’ll calculate are the vertical components of these two displacement
vectors.
Let’s begin with the quantity ℎ
two, the height of the steeper-climbing aircraft. We can identify this right-angled
triangle in the diagram. The hypotenuse of the triangle is
the path that the steeper-climbing aircraft moves along. This horizontal side has a length
of 2500 meters. That’s the horizontal distance
moved by the aircraft. And this vertical side has a length
of ℎ two. That’s the height that this
aircraft has climbed when it has traveled 2500 meters horizontally.
We know that the angle that the
steeper-climbing aircraft makes to the ground is 45 degrees. That’s the value of this angle in
the bottom-left corner of the triangle. To work out ℎ two, we need to
recall a useful trigonometric equation.
Let’s consider a general
right-angled triangle and suppose that this angle is 𝜃. We’ll label the length of the side
opposite this angle as 𝑜 and the length of the side adjacent to it as 𝑎. Then, for this general right-angled
triangle, tan 𝜃 is equal to 𝑜 divided by 𝑎. If we compare this general
right-angled triangle with the one that we’ve identified in our diagram, then we can
see precisely why this equation is going to be useful to us.
The equation connects the angle 𝜃,
the length of the side adjacent to the angle, and the length of the side opposite
it. That means that if we know any two
of these quantities, we can use this equation to calculate the third. In the triangle in this diagram, we
know this angle is 45 degrees, and so that’s our value for the quantity 𝜃. We also know that the length of the
side adjacent to this angle is 2500 meters, so that’s our value of 𝑎. This vertical side of the triangle
opposite the 45-degree angle has a length ℎ two that we’re trying to find. This is our value for the quantity
𝑜.
So then we have values for both 𝜃
and 𝑎, and we want to work out 𝑜. So let’s rearrange this equation to
make 𝑜 the subject. To do this, we first need to
multiply both sides of the equation by 𝑎. Then, canceling the 𝑎’s on the
right-hand side, we have 𝑜 equals 𝑎 multiplied by tan 𝜃. Substituting in our values for 𝑎
and 𝜃 and replacing 𝑜 by ℎ two, we have that ℎ two equals 2500 meters multiplied
by tan 45 degrees. Now, tan 45 degrees is equal to
one. So we find that ℎ two is simply
equal to 2500 meters.
Now that we’ve worked out the value
of ℎ two, let’s clear some space and move on to finding the value of ℎ one. To find ℎ one, we need to identify
a second right-angled triangle in the diagram. Specifically, that’s this triangle
shown in pink. The hypotenuse of this triangle is
the path of the less steeply climbing aircraft. The bottom-left angle in this
triangle is 20 degrees, since that’s the angle that this aircraft makes to the
ground.
Both these two aircraft travel the
same horizontal distance. And so this triangle has a
horizontal side length of 2500 meters, just like the first one did. The vertical side of this triangle
is the height ℎ one that this less steeply climbing aircraft reaches. So, in this case, the value of 𝜃
is 20 degrees, the adjacent side 𝑎 is 2500 meters, and the opposite side 𝑜 is ℎ
one.
Substituting these values into our
equation, we have that ℎ one is equal to 2500 meters multiplied by tan 20
degrees. Evaluating this expression, we find
that ℎ one is equal to 909.9256 et cetera meters.
Now that we’ve worked out both ℎ
two and ℎ one, we can subtract ℎ one from ℎ two to find this height here. ℎ two minus ℎ one is equal to 2500
meters minus 909.9256 meters, which works out as 1590.074 meters. We’re told to give our answer to
the nearest meter. Rounding our result, we then get
our final answer. To the nearest meter, ℎ two is 1590
meters vertically above ℎ one.
