Three points P, Q and R are on level ground. Q is 240 m from P on bearing of 230°. R is 120 m to the east of P.
(b) Determine
(i) The distance R from Q;
(ii) The bearing of R form Q.
(c) A vertical post stands at P and another one at Q. A bird takes 18 seconds to fly directly from the top of the post at Q to the pot of the post at P. Given that the angle of depression of the top of the post at P from the top of post at Q is 9°, calculate:
(i) The distance to the nearest meter, the bird covers;
(ii) The speed of the bird in km/h
still posting this, with no input of your own?
using the law of cosines,
RQ^2 = 240^2 + 120^2 - 2*240*120*cos130°
RQ = 330.19
To find the bearing of R from Q, the displacement is
y = 240 sin40° = 154.26
x = 240 cos40° + 120 = 303.85
So, QR make an angle of
arctan(y/x) = 27° with the x-axis.
So, the bearing of R from Q is 90-27 = 63°
Since PQ = 240, the line from the top of post P to the top of post Q is
d = 240/cos9° = 243
its speed is 243/18 = 13.5 m/s
Double-check my math, and next time, how about including some of your work? You can't be totally clueless ...
To solve this problem, we will use trigonometry and geometric concepts. Let's break down each part of the question step by step.
(b) (i) To determine the distance R from Q, we can use the concept of right-angled triangles.
Since Q is 240 m from P on a bearing of 230°, we can draw a diagram to visualize the situation.
```
P□---------------------□ Q
|\
| \
| \
| \
| \
| \
| \
R | \
```
The distance R from P to Q forms the hypotenuse of a right-angled triangle. The angle at Q can be found as 180° - 230° = 50° because the sum of the angles in a triangle is 180 degrees.
Using trigonometry, we can find the distance R using the sine function: sin(angle) = opposite/hypotenuse.
sin(50°) = R/240
R = 240 * sin(50°)
R ≈ 183.72 m
Therefore, the distance R from Q to P is approximately 183.72 m.
(ii) To determine the bearing of R from Q, we use the concept of bearings, where the bearing is measured clockwise from the north direction.
In this case, R is 120 m to the east of P, which means it is on a bearing of 90°. However, we need to adjust this bearing by the angle at Q (which is 50°) because we are looking for the bearing from Q.
Therefore, the bearing of R from Q is 90° + 50° = 140°.
Hence, the bearing of R from Q is 140°.
(c) (i) To calculate the distance the bird covers, we can use the concept of angle of depression and trigonometry.
Given that the angle of depression from the top of the post at Q to the top of the post at P is 9°, we can draw a right-angled triangle to represent this situation.
```
Q───────────□───P
|\
| \
| \
| \
| \
| \
| \
| \
|________\
```
The horizontal distance from the top of the post at Q to P is the adjacent side of the right-angled triangle, and the vertical distance (height) from the top of the post at Q to P is the opposite side. The bird's path will be the hypotenuse of this triangle.
Using trigonometry, we can find the horizontal distance (adjacent) covered by the bird using the tangent function: tan(angle) = opposite/adjacent.
tan(9°) = opposite/adjacent
adjacent = opposite / tan(9°)
adjacent ≈ height / tan(9°)
Since we are not given the height (opposite), we cannot calculate the exact distance covered by the bird without knowing the height.
(ii) To calculate the speed of the bird in km/h, we would need the distance covered by the bird and the time it took.
Since we don't have the exact distance covered by the bird, we cannot calculate its speed.
In conclusion, we were able to solve parts of the question, but were unable to calculate the exact distance covered by the bird or its speed without additional information.