A temporary support cable for a radio antenna is 110m long and has an angle of elevation of 30 degrees. Two other support cables are already attached, each at an angle of elevation of 70 degrees. How long to the nearest metre, is each of the shorter cables?
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Please refer to the attached figure (Figure 1 shows the given).
The triangle formed by the ground, the antenna and the 110-meter support cable is a 30-60-90 triangle. The cable support is the hypotenuse while the hight of the antenna is the shorter leg (since it is opposite the 30-degree angle). The shorter leg's length is half that of the hypotenuse: 55 meters. The longer leg is `sqrt(3)` times the shorter leg: `55 sqrt(3)` meters.
Use the definition of sine for the shorter cable: `sin(theta) = (opp)/(hypo)`
Opposite here refers to the antenna (55-meters from the ground) while hypotenuse refers to the length of the shorter cable. Isolating the hypotenuse: `hypo = (opp)sin(theta) = (55m)/(sin 70)=58.5`
Hence, assuming that the shorter cables are of the same length, they both have a length of 59 meters (to the nearest meters).
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