from an observation point A, a fire is spotted at a bearing of 62 degrees. the same fire is spotted from an observation point B, 42 miles due east from A, at a bearing of 332 degrees. how far is observation point B from the fire?
Bearing is the angle (0 to 360°) measured clockwise from the north.
We have a triange ABF whose base AB is 42 miles, angle A is 90-62° from due East, and angle B is 332-270 from due west.
Solve triangle ABF and calculate BF
To find the distance between observation point B and the fire, we can use trigonometric functions and the given information about the bearings and distances.
Step 1: Draw a diagram
First, draw a diagram with observation point A, observation point B, and the location of the fire. Label the distance between A and B as 42 miles and the bearing of the fire from A as 62 degrees.
Step 2: Calculate the internal angle
Since the bearing of the fire from observation point A is 62 degrees, we can determine the internal angle at observation point A:
Internal angle at A = 180 degrees - 62 degrees = 118 degrees
Step 3: Calculate the internal angle at B
Since the bearing of the fire from observation point B is 332 degrees, we can determine the internal angle at observation point B:
Internal angle at B = 180 degrees - 332 degrees = -152 degrees
Step 4: Convert the negative angle to positive
Since we need a positive angle for our calculations, add 360 degrees to the negative angle:
Internal angle at B = -152 degrees + 360 degrees = 208 degrees
Step 5: Use the Law of Sines
We can now use the Law of Sines to find the distance between observation point B and the fire. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In this case, we have the following equation:
sin(118 degrees) / 42 miles = sin(208 degrees) / x
Where x represents the distance between observation point B and the fire.
Step 6: Find x
Solve the equation for x:
x = (42 miles * sin(208 degrees)) / sin(118 degrees)
Using a scientific calculator, calculate the value of sin(208 degrees) and sin(118 degrees). Then substitute the values into the equation to find x.
Step 7: Calculate the final value of x
Compute the value of x to find the distance between observation point B and the fire.
Following these steps, you should be able to find the distance between observation point B and the fire.