Question

A park ranger at point A observes a fire in the direction N 25°36'E. Another ranger at point B, 5 miles due east of A, sites the same fire at N 56°19'W. Determine the distance from point B to the fire. Round answer to two decimal places.

Answer 1

To determine the distance from point B to the fire, construct a triangle from the given angles using the bearings provided, and then apply the law of sines. This will provide the distance between point B and the fire after performing the necessary calculations.

Step-by-step explanation:

The problem is a classic example of trigonometry and triangle problem solving techniques. To determine the distance from point B to the fire, we need to construct a triangle with the given bearings and distances. Given that ranger A observes the fire at N 25°36'E and ranger B, who is 5 miles due east from A, observes the fire at N 56°19'W, we can set up a triangle where we have one side (the 5 miles between rangers A and B) and two non-adjacent angles.

Since there are 90 degrees between the cardinal directions, we can determine the internal angles of our triangle by subtracting the ranger's bearings from 90 degrees:

Angle at A (western angle) = 90° - 25°36' = 64°24'

Angle at B (eastern angle) = 90° - 56°19' = 33°41'

With these angles and the side between point A and B known, we can use the law of sines to find the distance from point B to the fire. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Let x be the distance from B to the fire, we get:

(sin(64°24') / 5) = (sin(33°41') / x)

Thus, x = 5 * (sin(33°41') / sin(64°24')). Calculate this using a calculator to get the distance from point B to the fire.

Answer 2

Using the law of sines with the angles provided, the distance from point B to the fire is calculated to be approximately 29.71 miles.

Step-by-step explanation:

To determine the distance from point B to the fire spotted by the park rangers, we can use the law of sines in a triangle. Given that ranger at point A observes the fire in the direction N 25°36'E and another ranger at point B, which is 5 miles due east of A, sights the same fire at N 56°19'W, we can form a triangle with the given angles and one known side.

First, we need to find the angle at the fire's location. Since point A observes the fire to the northeast and point B observes it to the northwest, the internal angle at the fire's location is the sum of 25°36' and 56°19', plus the 90 degrees between north and east. This sum gives us 171°55'. Therefore, the angle opposite the known side (5 miles between points A and B) is 180° - 171°55' = 8°5'.

Using the law of sines, the formula for the distance from point B to the fire (let's call this side c) will be:
c/sin(56°19') = 5 miles/sin(8°5'). Solving for c will give us the distance.

Let's calculate this:

sin(56°19') = sin(56 + 19/60) ≈ 0.833

sin(8°5') = sin(8 + 5/60) ≈ 0.140

c = 5 miles * (sin(56°19') / sin(8°5')) ≈ 5 miles * (0.833 / 0.140) = 29.71 miles

The approximate distance from point B to the fire, rounded to two decimal places, is 29.71 miles.