The distance between the fire towers and the fire is an illustration of bearing and distance
The distance d of the fire from segment AB is 24.76 kilometers
The given parameters are:
\(AB = 27\)
\(\theta = 70^o\)
\(\omega = 54^o\)
Calculate distance d (see attachment) using tangent ratio.
Considering triangle BCD, we have:
\(\tan(54) = \frac{d}{27 - x}\)
Make d the subject
\(d = (27 - x)\times \tan(54)\)
Considering triangle ACD, we have:
\(\tan(70) = \frac{d}{x}\)
Make d the subject
\(d =x \times \tan(70)\)
Substitute \(d =x \times \tan(70)\) in \(d = (27 - x)\times \tan(54)\)
\(x \times \tan(70) = (27 - x) \times \tan(54)\)
\(x \times 2.7475 = (27 - x) \times 1.3764\)
\(2.7475x = (27 - x) \times 1.3764\)
Open brackets
\(2.7475x = 37.1628 - 1.3764x\)
Collect like terms
\(2.7475x +1.3764x= 37.1628\)
\(4.1239x= 37.1628\)
Divide both sides by 4.1239
\(x= 9.0112\)
Substitute \(x= 9.0112\) in \(d =x \times \tan(70)\)
\(d = 9.0112 \times \tan(70)\)
\(d = 24.76\)
Hence, the distance d of the fire from segment AB is 24.76 kilometers
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