P and Q are two observation post on the same horizontal ground as the foot R of a vertical pole RT of height 12m. From R, the bearing of P and Q are 195 and 225 degrees respectively, the angles of elevation of T from P and Q are 35 and 21 degrees respectively. Calculate the distance of PQ and the bearing of P from Q
QR/12 = cot21°
PR/12 = cot35°
PQ^2 = QR^2 + PR^2 - 2*QR*PR*cos30°
So, now we know that
QR = 31.26, PR = 17.14, PQ = 18.52
So, the bearing of P from Q is 162.53°
Well, well, well, it seems like we have a bit of a geometry puzzle here. Don't worry, I've got my funny hat on and I'm ready to clown around with some math!
Let's start by visualizing the situation. We have two observation posts P and Q, and a jolly old pole RT sticking up in the air. Now, the distance we're after is the distance between P and Q, so let's call that distance "x." Got it? Great!
Now, let's look at the angles of elevation. From P, the angle of elevation to the top of the pole is 35 degrees. So if we draw a line from P to the top of the pole, we'll have a right-angled triangle with one angle of 35 degrees. And we also know that the height of the pole is 12 meters. So, using some trigonometry magic, we can find the length of RP, which I'll call "y."
Using the tangent function, we can say that tan(35 degrees) = 12/y. Now, if we rearrange this equation a bit, we get y = 12/tan(35 degrees). Now go ahead and grab your calculator, my friend, and calculate y for me.
*Calculator sound effects*
Ah, marvelous! Now we know the length of RP. But wait, there's more math to be had! We need to find the length of RQ, which I'll call "z."
From Q, the angle of elevation to the top of the pole is 21 degrees, so we can set up a nice little equation just like before. Tan(21 degrees) = 12/z. Rearrange it, calculate it, and voila, you've got the length of RQ!
Now, to find the distance PQ, we can use a bit of trigonometry trickery once again. We'll use the sine rule this time. We know that the angle R is 360 minus the bearing of P from Q, which is 225 degrees. So R = 360 - 225 = 135 degrees. Now, we can set up the sine rule:
PQ/sin(R) = RP/sin(35) = RQ/sin(21).
Plug in the values you've calculated and solve for PQ. I'll let you do the legwork on that one, my friend.
And finally, the bearing of P from Q. Well, since we're talking about bearings here, we'll use a bit of compass lingo. The bearing is basically the direction from one point to another, measured in degrees clockwise from north.
To find the bearing of P from Q, we need to calculate the angle between the line PQ and the north direction. But you see, my friend, this angle will be the sum of the bearing of Q from north (225 degrees) and the bearing of P from Q. So, add them up, adjust if necessary to keep it within 360 degrees, and there you have it – the magnificent bearing of P from Q!
Now, grab that pencil, put on your clown nose, and let's crunch some numbers!
To solve this problem, we can use trigonometry and geometry principles.
1. Distance PQ:
Let's draw a diagram to visualize the problem.
We have the following information:
- The height of the pole, RT, is 12m.
- The angles of elevation of T from P and Q are 35° and 21°, respectively.
- The bearing of P and Q from R are 195° and 225°, respectively.
Based on the angles of elevation, we can determine that triangle PRT and triangle QRT are right-angled triangles.
To find the distance PQ, we need to find the lengths of PR and QR.
Using trigonometry, we can start with triangle PRT:
- tan(35°) = PR/RT
- PR = RT * tan(35°)
- PR = 12m * tan(35°)
- PR ≈ 8.01895m
Similarly, using triangle QRT:
- tan (21°) = QR/RT
- QR = RT * tan(21°)
- QR = 12m * tan(21°)
- QR ≈ 4.62194m
The distance PQ is the sum of PR and QR:
- PQ = PR + QR
- PQ ≈ 8.01895m + 4.62194m
- PQ ≈ 12.6409m
Therefore, the distance of PQ is approximately 12.6409 meters.
2. Bearing of P from Q:
To find the bearing of P from Q, we need to find the angle formed by the line segment PQ and the North direction.
The bearing of P from R is given as 195°, which can be considered as the bearing from R to P, clockwise from the North direction.
Similarly, the bearing of Q from R is given as 225°.
To find the bearing of P from Q, we can subtract the bearing of Q from the bearing of P, clockwise:
- Bearing of P from Q = Bearing of P from R - Bearing of Q from R
- Bearing of P from Q = 195° - 225°
- Bearing of P from Q = -30°
Since the bearing is negative, it means that the angle is taken clockwise from the South direction.
Therefore, the bearing of P from Q is 30° (clockwise) from the South direction.
To find the distance of PQ, we can use trigonometry and the concept of right triangles.
First, let's draw a diagram to visualize the problem:
P
/\
/ \
/ \
/ \
/ \
R------------Q
\ /
\ /
\ /
\ /
\/
From the diagram, we can see that we have two right triangles: PTR and QTQ'.
Let's find the lengths of PT and QT:
In triangle PTR, we have the opposite side (height of the pole, RT) and an angle of elevation (35 degrees). We can use the tangent function to find PT:
tan(35 degrees) = RT / PT
tan(35 degrees) = 12 / PT
To find PT, rearrange the equation:
PT = 12 / tan(35 degrees)
Using a calculator, we get PT ≈ 16.85 m.
Similarly, in triangle QTQ', we have the opposite side (height of the pole, RT) and an angle of elevation (21 degrees). We can use the tangent function to find QT:
tan(21 degrees) = RT / QT
tan(21 degrees) = 12 / QT
To find QT, rearrange the equation:
QT = 12 / tan(21 degrees)
Using a calculator, we get QT ≈ 20.8 m.
Now that we have the lengths of PT and QT, we can find the distance PQ by adding them together:
PQ = PT + QT ≈ 16.85 m + 20.8 m ≈ 37.65 m
Therefore, the distance of PQ is approximately 37.65 meters.
To find the bearing of P from Q, we can use the concept of bearings. Bearings are measured clockwise from the north direction.
From the diagram, we can see that the bearing of Q from R is 225 degrees. To find the bearing of P from Q, we can subtract this angle from 360 degrees:
Bearing of P from Q = 360 degrees - bearing of Q from R
Bearing of P from Q = 360 degrees - 225 degrees
Bearing of P from Q = 135 degrees
Therefore, the bearing of P from Q is 135 degrees.