13.4 meters
Step-by-step explanation:
About the diagram
Attached is a diagram of the problem. The directions North and East are shown so that the usual clockwise from +x measurement of angles will correspond to the bearing angles given in the problem. That is, the bearing of 120° is an angle of 120° measured from North toward East. (The mirroring across the line y=x can be a little mind-bending, but it is isomorphic, so all angles and lengths remain unchanged.)
The other feature of this diagram is the projection of the 3-D problem into two dimensions. Effectively, the center angle rabbit-O-coyote represents a plan view (in the plane of the ground), and the triangles coyote-O-C1 and rabbit-O-R1 represent side views (views in the vertical plane).
The side views let us work out the ground distances O-rabbit and O-coyote, so that we can find the ground distance rabbit-coyote as the problem requests.
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Problem solution
The distance from the tree (O) to the rabbit is the "adjacent" leg of the 30° angle of elevation from the rabbit to the tree top. The 10 m tree height is the "opposite" leg of that rabbit-O-R1 right triangle. We know the ratio of opposite to adjacent sides gives the tangent of the angle, so we have ...
tan(30°) = 10/(rabbit distance from tree)
Solving for the rabbit distance, we get
rabbit distance = 10/tan(30°) ≈ 17.3205 m
Similarly, the coyote distance from the tree will be ...
coyote distance = 10/tan(20°) ≈ 27.4748 m
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These are two legs of the rabbit-O-coyote triangle. The angle at O between the rabbit and coyote is 120° -97° = 23°. These values are sufficient to let us use the Law of Cosines to find the distance d from rabbit to coyote:
d² = r² +c² -2rc·cos(23°)
d² = 17.3205² +27.4748² -2·17.3205·27.4748·cos(23°) ≈ 178.769
d ≈ √178.769 ≈ 13.37 . . . . meters
The distance from the rabbit to the coyote is about 13.4 meters.
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Additional comment
Note that the angle of depression to the rabbit from the horizontal is the same as the angle of elevation from the rabbit. This lets us draw the diagram without a bunch of extra lines.
