Surveying
A surveyor wishes to find the distance across a swamp. The bearing from A to B (Segment AB is opposite side of triangle) is N 32° W. The surveyor walks 50 meters from A to C, and at the point C the bearing to B is N 68° W. (Segment AC is adjacent side of triangle & Segment BC is hypotenuse side of triangle.)
A) Find bearing from A to C?
B) What is the distance from A to B?
Confusing description. Where does the swamp come in ?
After a few tries I got a right-angled triangle with angle A = 90°
I sketched a vertical line AP .
From your description, angle BAP = 32°, leaving
angle PAC = 58°
I also drew a vertical from C to Q
so that angle QCB = 68°
let angle BCA = Ø
since AP and CQ are parallel
angle PAC + angle ACQ = 180°
58 + Ø+68=180
Ø = 54°
OK then....
A) bearing from A to C is N 58° E
B) tan 54° = AB/50
AB = 50tan54 = 68.82 m
Reiny thanks a lot for the help! I welcome you with a nice smile! Here is the website where I can show how the problems look/appear like...
Google this:
4.8 applications and models Comcast
Click the first result that appears on the web search page.
Go to page 8 in the PDF document on problem #33.
A) Well, if the surveyor is walking towards the west, then he must be taking a detour to avoid stepping on any snakes in the swamp! The bearing from A to C would be S 32° E, so he's going in the opposite direction.
B) As for the distance from A to B, let's just hope the surveyor didn't sink in the swamp! To find the distance, we can use some trigonometry. If segment AC is adjacent and segment BC is the hypotenuse of a right triangle, we can use the cosine function.
cos(32°) = Adjacent / Hypotenuse
cos(32°) = 50 meters / AB
Now, we can solve for AB:
AB = 50 meters / cos(32°)
And that's a math problem, not a joke!
To find the bearing from A to C, we need to calculate the difference in angles between the bearing from A to B and the bearing from C to B.
- The bearing from A to B is N 32° W.
- The bearing from C to B is N 68° W.
A) Bearing from A to C:
To find the bearing from A to C, we subtract the bearing from C to B from the bearing from A to B.
Bearing from A to C = (Bearing from A to B) - (Bearing from C to B)
Bearing from A to C = N 32° W - N 68° W
Bearing from A to C = N (32° - 68°) W
Bearing from A to C = N (-36°) W
Bearing from A to C = N 36° E
So, the bearing from A to C is N 36° E.
B) Distance from A to B:
To find the distance from A to B, we can use the given information about the lengths of segments AC and BC. We can use the Law of Cosines to calculate the length of AB.
In triangle ABC, angle ACB = 180° - (angle ACB + angle CAB)
Angle ACB = N 68° W - N 32° W = N 36° W
Angle CAB = 180° - (68° + 36°) = 76°
Using the Law of Cosines, we have:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(CAB)
Substituting the given values:
AB^2 = 50^2 + BC^2 - 2 * 50 * BC * cos(76°)
Now we need to find the value of BC. BC is the hypotenuse of a right-angled triangle, and we can use trigonometry to find it.
cos(76°) = BC / AC
cos(76°) = BC / 50
Solving for BC:
BC = 50 * cos(76°)
Now we can substitute this value back into the equation for AB^2:
AB^2 = 50^2 + (50 * cos(76°))^2 - 2 * 50 * (50 * cos(76°)) * cos(76°)
AB = √(AB^2)
Calculate AB using the above formula, and you will find the distance from A to B.
To answer these questions, we can use trigonometric concepts and the information given in the problem. Specifically, we will use the concept of bearing angles and the law of sines.
A) Finding the bearing from A to C:
1. Draw a triangle ABC, where A and C are the points mentioned in the problem and B is the point opposite to segment AB.
2. According to the problem, the bearing from A to B is N 32° W. This means the angle BAC is 32°.
3. Since we want to find the bearing from A to C, we need to find the angle BCA.
4. The sum of angles in a triangle is always 180°, so we can calculate angle BCA by subtracting the known angles from 180°.
BCA = 180° - BAC - BCA
BCA = 180° - 32° - ???
BCA = ???
B) Finding the distance from A to B:
1. To find the distance from A to B, we can use the law of sines, which relates the ratios of the lengths of the sides of a triangle to the sines of its angles.
2. In triangle ABC, we know the following information:
- Angle BAC = 32°
- Side AC = 50 meters (given length)
- Angle BCA (which we calculated in part A)
- Side AB (the distance we want to find)
3. Using the law of sines, we can set up the following equation:
sin(BAC) / AB = sin(BCA) / AC
sin(32°) / AB = sin(BCA) / 50
Now, we need to solve for AB.
By following these steps and solving the equations, you will be able to find the bearing from A to C and the distance from A to B.