Question

Tara wants to fix the location of a mountain by taking measurements from

two positions 3 miles apart. From the first position, the angle between
the mountain and the second position is 78°. From the second position,
the angle between the mountain and the first position is 53º. Find the
distance from each of the positions to the mountain. Round your
answers to the nearest tenth of a mile.

Answer 1

Distance from point 1 = 3.2 miles

Distance from point 2 = 3.9 miles

Explanation:

Angle of elevation of the mountain at point 3 from point '1' is 78° and angle of elevation from point '2' is 53°.

m∠1 + m2 + a = 180°

53° + 78° + a° = 180°

a = 180 - 131

a = 49°

By applying Sine rule in the given triangle,

`(SinA)/(a)=(SinB)/(b)=(SinC)/(c)`

`\frac{\text{Sin}49}{3}=\frac{\text{Sin}53}{b}=\frac{\text{Sin}78}{c}`

`\frac{\text{Sin}49}{3}=\frac{\text{Sin}53}{b}`

b =
`\frac{3\text{Sin}53}{\text{Sin}49}` = 3.17

≈ 3.2 miles

Similarly,
`\frac{\text{Sin}49}{3}=\frac{\text{Sin}78}{c}`

c =
`\frac{3\text{Sin}78}{\text{Sin}49}`

c = 3.88 ≈ 3.9 miles

Distance of the mountain from the points 1 = 3.2 miles

Distance of the mountain from the points 2 = 3.9 miles