Video Transcript
𝐸 and 𝐴 are two hot-air balloons
flying at heights of 117 meters and 84 meters, respectively. The angles of depression at a point
𝐶 on the ground from 𝐸 and 𝐴 are 40 degrees and 29 degrees, respectively. Find the distance between the
balloons giving the answer to the nearest meter.
Let’s start by identifying what we
have and what we need. We have two right triangles with a
side and an angle each, which means we can calculate the additional sides if
necessary. We also have a nonright triangle,
triangle 𝐴𝐶𝐸. And the distance between 𝐴 and 𝐸
is what we’re trying to find. We’re trying to find the length of
one of the sides of a nonright triangle.
When working with nonright
triangles, we know that we’ll be using either the sine rule or the cosine rule to
find the value of the side labeled 𝑥. In order to do this, however, we
need to find some additional information. First, we see that the line 𝐷𝐵
representing the ground is a straight line. And we know that angles in a
straight line add to 180 degrees. If we subtract 40 and 29 from 180,
we find the value of the angle that we’ve labeled 𝜃. It’s 111 degrees.
There’s still quite a bit of
information we need to solve for the length of 𝑥. Let’s sketch separately the right
triangle from our diagram to help us see what we can do next. It’s a right triangle with one
angle and one side. This means we can use trigonometry
to find the additional sides. The hypotenuse of this right
triangle shares a side with the larger triangle 𝐸𝐶𝐴, which we’re trying to find a
missing value for. To find the hypotenuse, we can use
the sine relationship, where sin of 𝜃 is equal to the opposite over the
hypotenuse. This means for the smaller right
triangle, sin of 40 degrees will be equal to 117 over 𝐸𝐶.
Multiplying both sides of the
relationship by 𝐸𝐶 gives us 𝐸𝐶 times sin of 40 degrees equals 117. And then dividing through by sin of
40 degrees gives us 𝐸𝐶 equals 117 over sin of 40 degrees. 𝐸𝐶 is equal to 182.01
continuing. You’ll notice here that we’re not
going to round. Instead, we’ll use this exact value
and save our rounding for the final step. This reduces the chance of error
due to early rounding.
Let’s repeat this process with the
smaller right triangle 𝐴𝐵𝐶, where the hypotenuse side 𝐴𝐶 is shared with the
triangle 𝐴𝐶𝐸. Again, we know the opposite side
length that we’re looking for the hypotenuse, which means we’ll use the sine
relationship. sin of 29 degrees equals 84 over
𝐴𝐶. Multiplying both sides by 𝐴𝐶,
𝐴𝐶 times sin of 29 degrees equals 84. Dividing through by sin of 29
degrees, we find that 𝐴𝐶 equals 84 over sin of 29 degrees. 𝐴𝐶 is equal to 173.26
continuing.
Looking back at our diagram, we now
have a nonright triangle in which we know two side lengths and a shared angle. Under these conditions, we can
solve for that third side. This means to find the value of 𝑥,
we can use the law of cosines, which tells us 𝑐 squared equals 𝑎 squared plus 𝑏
squared minus two 𝑎𝑏 times cos of 𝑐, where 𝑐 is the side length we’re trying to
find. And we know the other two sides and
the angle opposite the side you’re trying to find.
Into this formula we can substitute
𝑒 in place of 𝑏 because we’re working with triangle 𝐴𝐸𝐶 instead of triangle
𝐴𝐵𝐶, where lowercase 𝑒 is the side opposite angle 𝐸, lowercase 𝑎 the side
opposite angle 𝐴, and lowercase 𝑐 the side opposite angle 𝐶. Substituting these values in gives
us 𝑥 squared equals 182.01 continuing squared plus 173.26 continuing squared minus
two times 182.01 continuing times 173.26 continuing times cos of 111 degrees.
Remember when you’re entering these
into your calculator to use the saved values of 𝐴 and 𝐸. This maintains the accuracy of our
calculations. 𝑥 squared is then equal 85,620.14
continuing. Taking the square root of both
sides will give us 𝑥 is equal to 292.6 continuing. Remember, we can disregard the
negative square root here, as we’re dealing with distance. And 292.6 rounded to the nearest
meter is 293. In the context of our question,
this means that the hot-air balloons 𝐴 and 𝐸 are 293 meters apart.
