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 78o. From the second position, the angle between the mountain and the first position is 53o. Find the SHORTEST distance to the mountain. Round to the nearest hundredth of a mile (two decimal places).

Answer 1

Shortest distance from the mountain is 3.17 miles.

Explanation:

From the figure attached,

Let a mountain is located at point A.

Angle between the mountain and point B (∠B) = 53°

Angle between the mountain and point C (∠C) = 78°

Distance between these points = 3 miles

Since, m∠A + m∠B + m∠C = 180°

m∠A + 53° + 78° = 180°

m∠A = 180°- 131° = 49°

By applying sine rule in triangle ABC,

`\frac{\text{sin}(49)}{BC}=\frac{\text{sin}(53)}{AC}= \frac{\text{sin}(78)}{AB}`

`\frac{\text{sin}(49)}{3}=\frac{\text{sin}(53)}{AC}= \frac{\text{sin}(78)}{AB}`

`\frac{\text{sin}(49)}{3}=\frac{\text{sin}(53)}{AC}`

AC =
`\frac{3\text{sin}(53)}{\text{sin}(49)}`

AC = 3.17 miles

`\frac{\text{sin}(49)}{3}=\frac{\text{sin}(78)}{AB}`

AB =
`\frac{3\text{sin}(78)}{\text{sin}(49)}`

AB = 3.89 miles

Therefore, shortest distance from the mountain is 3.17 miles.