Wes saw fireworks at 28 degrees; Abby (who was standing 20 m away from him) saw the same fireworks at a 40 degree angle. Where was Abby standing?
Wes saw fireworks at 28 degrees; Abby (who was standing 20 m away from him) saw the same fireworks at a 40 degree angle. Where was Abby standing?
H is at height of explosion above ground
B is point below explosion on ground
first find distance AH
other angle at A = angle HAW = 180 - 40 = 140
so third angle in that triangle WHA = 180-140-28=12 degrees
so
sin 12/20 = sin 28/AH
AH = 20 sin 28/sin 12 = 20 * .426 / .187 = 45.6
that is thehypotenuse of triangle ABH
so
sin 40 = BH / 45.6
and
cos 40 = AB / 45.6
To find out where Abby was standing, we need to use trigonometry and the concept of angles.
Let's assume that the position where Wes saw the fireworks is point A, and Abby's position is point B. We are given the angle at which Wes saw the fireworks (28 degrees) and the angle at which Abby saw the fireworks (40 degrees).
Now, let's draw a diagram to visualize the scenario:
B
|\
| \
| \
| \
| \ 20 m
| \
| \
| \
A -----C--------->
Wes 20 m
In this diagram, AB represents the distance between Wes and Abby (which is given as 20 m). We want to find the position of point B.
To solve this problem, we need to use the tangent function, which relates the angle of a right triangle to the ratio of the lengths of its sides.
The tangent of an angle is defined as the opposite side divided by the adjacent side. In this case, AB is the opposite side and BC is the adjacent side.
So, we can write the equation:
tan(28 degrees) = opposite/adjacent
tan(28 degrees) = AB/20
To find the value of AB, we rearrange the equation:
AB = tan(28 degrees) * 20
We can use a scientific calculator or an online trigonometric calculator to find the value of the tangent of 28 degrees and then multiply it by 20 to get the value of AB.
AB ≈ 10.558 m
So, Abby was standing approximately 10.558 meters away from Wes, on the side opposite to the fireworks.