Question

A small plane and a large plane are 6.8km from each other, at the same altitude (height). From an observation tower, the two airplanes are separated by an angle of 58°. The large plane is 5.2km from the observation tower. a. Draw a diagram to represent this situation. b. How far is the small plane from the observation tower, to the nearest tenth of a kilometer?

Answer 1

7.9km

Explanation:

(a)See attached for the diagram representing this situation.

(b)

In Triangle ABC

`\text{Using Law of Sines}\\(\sin A)/(a)=(\sin C)/(c) \\(\sin A)/(5.2)=(\sin 58^\circ)/(6.8) \\\sin A=5.2 * (\sin 58^\circ)/(6.8)\\A=\arcsin (5.2 * (\sin 58^\circ)/(6.8))\\A=40.43^\circ`

Next, we determine the value of Angle B.

`\angle A+\angle B+\angle C=180^\circ\\40.43+58+\angle B=180^\circ\\\angle B=180^\circ-(40.43+58)\\\angle B=81.57^\circ`

Finally, we find b.

`\text{Using Law of SInes}\\(b)/(\sin B)=(c)/(\sin C) \\(b)/(\sin 81.57^\circ)=(6.8)/(\sin 58^\circ) \\b=(6.8)/(\sin 58^\circ) * \sin 81.57^\circ\\b=7.9km $ (to the nearest tenth of a kilometer)`

The distance between the small plane and the observation tower is 7.9km.