### Video Transcript

๐ธ and ๐ด are two hot-air balloons
flying at heights of 117 meters and 84 meters, respectively. The angles of depression at a point
๐ถ on the ground from ๐ธ and ๐ด are 40 degrees and 29 degrees, respectively. Find the distance between the
balloons giving the answer to the nearest meter.

Letโs 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 ๐ด๐ถ๐ธ. And the distance between ๐ด and ๐ธ
is what weโre trying to find. Weโre trying to find the length of
one of the sides of a nonright triangle.

When working with nonright
triangles, we know that weโll be using either the sine rule or the cosine rule to
find the value of the side labeled ๐ฅ. In order to do this, however, we
need to find some additional information. First, we see that the line ๐ท๐ต
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 weโve labeled ๐. Itโs 111 degrees.

Thereโs still quite a bit of
information we need to solve for the length of ๐ฅ. Letโs sketch separately the right
triangle from our diagram to help us see what we can do next. Itโs 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 ๐ธ๐ถ๐ด, which weโre trying to find a
missing value for. To find the hypotenuse, we can use
the sine relationship, where sin of ๐ 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 ๐ธ๐ถ.

Multiplying both sides of the
relationship by ๐ธ๐ถ gives us ๐ธ๐ถ times sin of 40 degrees equals 117. And then dividing through by sin of
40 degrees gives us ๐ธ๐ถ equals 117 over sin of 40 degrees. ๐ธ๐ถ is equal to 182.01
continuing. Youโll notice here that weโre not
going to round. Instead, weโll use this exact value
and save our rounding for the final step. This reduces the chance of error
due to early rounding.

Letโs repeat this process with the
smaller right triangle ๐ด๐ต๐ถ, where the hypotenuse side ๐ด๐ถ is shared with the
triangle ๐ด๐ถ๐ธ. Again, we know the opposite side
length that weโre looking for the hypotenuse, which means weโll use the sine
relationship. sin of 29 degrees equals 84 over
๐ด๐ถ. Multiplying both sides by ๐ด๐ถ,
๐ด๐ถ times sin of 29 degrees equals 84. Dividing through by sin of 29
degrees, we find that ๐ด๐ถ equals 84 over sin of 29 degrees. ๐ด๐ถ 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 ๐ฅ,
we can use the law of cosines, which tells us ๐ squared equals ๐ squared plus ๐
squared minus two ๐๐ times cos of ๐, where ๐ is the side length weโre trying to
find. And we know the other two sides and
the angle opposite the side youโre trying to find.

Into this formula we can substitute
๐ in place of ๐ because weโre working with triangle ๐ด๐ธ๐ถ instead of triangle
๐ด๐ต๐ถ, where lowercase ๐ is the side opposite angle ๐ธ, lowercase ๐ the side
opposite angle ๐ด, and lowercase ๐ the side opposite angle ๐ถ. Substituting these values in gives
us ๐ฅ 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 youโre entering these
into your calculator to use the saved values of ๐ด and ๐ธ. This maintains the accuracy of our
calculations. ๐ฅ squared is then equal 85,620.14
continuing. Taking the square root of both
sides will give us ๐ฅ is equal to 292.6 continuing. Remember, we can disregard the
negative square root here, as weโre 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 ๐ด and ๐ธ are 293 meters apart.