From an observation point A, a forest ranger sights a fire in the direction S32°20'W (see the figure). From a point B, 5 miles due west of A, another ranger sights the same fire in the direction S57°40'E. Approximate, to the nearest tenth of a mile, the distance d of the fire from A.
You have two angles,
A=32°20'
B=57°40'
so, angle F at the fire, is 90°
d/5 = cosA = .845
d = 4.2 mi
If angle F had not been 90°, then using the law of sines,
d/sinA = 5/sinF
for this problem, sinF = 1
To solve this problem, we can use trigonometry and the information given about the angles and distances.
1. Let's start by drawing the triangle formed by points A, B, and the fire location F.
```
B
*------*
| /
| /
| /
| / F
|/
A
```
2. From the problem statement, we know that point B is 5 miles due west of point A.
3. We are given two angles:
- The angle at point A is S32°20'W.
- The angle at point B is S57°40'E.
4. To determine the distance of the fire from point A (let's call it d), we need to find the length of side AB.
5. First, let's find the length of side BA (the distance from B to A). Since we know that B is 5 miles directly west of A, side BA is simply 5 miles.
6. Now, we'll use trigonometry to find the length of side AF (the distance from A to the fire).
7. From the angle at point A, we know it is facing S32°20'W. This angle is the angle between side AF and the line due south from A (represented by a vertical line in our diagram).
8. We can find the length of side AF using the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, tan(32°20') = AF / BA
tan(32°20') = AF / 5
9. Rearranging the equation to solve for AF, we have AF = 5 * tan(32°20').
10. Using a calculator, evaluate tan(32°20') and multiply it by 5 to find the value of AF. Let's call this value X.
11. Now, let's move to point B. From the angle at point B, we know it is facing S57°40'E. This angle is the angle between side AF and the line due south from B (represented by a vertical line in our diagram).
12. We can find the length of side AF (which is the same as side BA) using the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, tan(57°40') = AF / BA
tan(57°40') = AF / 5
13. Rearranging the equation to solve for AF, we have AF = 5 * tan(57°40').
14. Using a calculator, evaluate tan(57°40') and multiply it by 5 to find the value of AF. Let's call this value Y.
15. Finally, the distance of the fire from point A (d) is the average of X and Y, since they represent the same distance AF.
To summarize:
- Calculate X = 5 * tan(32°20')
- Calculate Y = 5 * tan(57°40')
- Calculate d = (X + Y) / 2, rounded to the nearest tenth of a mile.
By following these steps, you should be able to approximate the distance d of the fire from point A.
To solve this problem, we can use trigonometry and the concept of triangulation.
Step 1: Draw a diagram to visualize the problem. Place point A at the observation point and point B, which is 5 miles due west of A. Label the direction of the fire as S32°20'W from A and S57°40'E from B.
Step 2: Observe that we have two angles provided: ∠A = 32°20' and ∠B = 57°40'. We can find the third angle, ∠C, using the fact that the sum of the angles in a triangle is 180°:
∠C = 180° - ∠A - ∠B
Step 3: Calculate ∠C using the values given:
∠C = 180° - 32°20' - 57°40'
= 180° - 90°
= 90°
Step 4: Use trigonometry to find the distance d. We have two right triangles: ΔABC and ΔDCB.
For triangle ΔABC:
AC = d (the distance from A to the fire)
BC = 5 miles (the distance from B to the fire)
For triangle ΔDCB:
∠DCB = 90°
DC = 5 miles (the distance from B to the fire)
∠DBC = 57°40'
Step 5: Use the tangent function to find the length of AC:
tan(∠DBC) = DC / BC
tan(57°40') = 5 / BC
BC = 5 / tan(57°40')
Step 6: Substitute the value of BC into the equation from triangle ΔABC:
AC = d = √(BC^2 + AB^2)
d = √((5 / tan(57°40'))^2 + 5^2)
Step 7: Calculate the value of d:
d ≈ √((5 / tan(57°40'))^2 + 5^2)
≈ √((5 / 1.541)^2 + 5^2)
≈ √((3.25)^2 + 5^2)
≈ √(10.5625 + 25)
≈ √35.5625
≈ 5.97
Approximate to the nearest tenth of a mile, the distance d of the fire from A is 5.97 miles.